Strong warmonger neighbour

Fight a Defensive War:

• Less Units needed doesn't mean no units. When fighting a defensive war, always station at least 1 ranged unit in the city, have a few others outside (behind) the city for extra defense if necessary. If it is a naval city station a ranged ship too. Have a fast-mover (cavalry) ready to kill siege units. Melee units are required to take a city, therefore target melee units appropriately (Though if it's mostly melee with some ranged, it might me best to remove the ranged since they do damage to the city without hurting themselves).
• Defensive Buildings & Improvements. Use Forts if available, a well-placed citadel can easily be a great defense. Walls, Armories, etc assist in increasing city-defense as well as having a stationed-unit. Certain policies benefit from garrisoned units as well (Tradition gains additional ranged-city-strength, Honor gains free maintenance, etc).
• Diplomacy. It might work just giving them some resources that you don't need for free. However, a strategy I like to do if I know someone is going to declare war on me is give them all my strategic resources, they build units using them, then declare war losing all those resources making the units that require them significantly weaker. (Note: this is a very situational strategy and probably won't work, though if you need that extra advantage it can be detrimental to your opponent).
• Other Civilizations. Try convincing another Civilization (i.e. Carthage) to go to war with Washington. Sometimes all it takes is a couple gold per turn or a Luxury item, especially if they dislike the civilization already. Worst case scenario see if they will trade you gold/luxury for going to war with Washington once the time comes.
• Gold. Use what little gold you have to get the ranged units you need. However this is for times of war and desperation. Don't waste your gold before the war begins or before you really need to.
• Pay attention. When you first see units moving toward you, you should already have 1-2 ranged units ready at that city. You should also be building more. If you start building immediately you should be able to get 1-2 more ranged units by the time they line your borders with their units (depending on production and terrain of course).
• Counter-Attack. If available get a few fast-movers (Cavalry typically) and send them to every enemy improvement. Pillage everything! I've found they eventually tend to surrender the war (occasionally yielding a city to you) when you do this. Don't take their cities unless you're able to deal with the warmonger penalty (which if you've only discovered Washington then it shouldn't matter really).

Expand Defensively: Ideally, I would put a city on the tile outlined in red. Why?

• River and Rough Terrain. The blue lines indicate where Enemy units will be stopped. It will become hard for these units to get to your city as quickly and therefore it will take longer. I've had situations where one more turn was all I needed.
• City Location (Tactics/Military). That's great and all, but why not settle on the tile highlighted in yellow? Well, if you did that, then the enemy could place his melee unit on the orange highlighted tile and fortify him. Allow it to fortify for two turns and you've made a big mistake. The 2-turn fortify bonus, the rough terrain bonus, and the heal per turn will be enough to keep your city distracted long enough for the ranged units to work it down so that the melee unit can capture it. They'll line their ranged units on the blue line and blitz you so fast you won't know what hit you.
• One exception: Ignore Line-Of-Sight. If you have a unit with ignore-line-of-sight promotion (via promotion or specific units like Artillery) then you can have a bit of an advantage with the rough terrain right next to your city in some circumstances. For example, if the enemy is sending in ranged units first you will be able to shoot them before they can shoot you (assuming they too don't have the promotion) and if they're a siege unit that adds even another shot you can take before them (due to setting up, technically 4 if you count the 2 shots from the city). Just something else to keep in mind.
• City Location (Strategy/Economy). So why not settle your city on the opposite side of the orange/red tile? Well this one isn't as bad but has a minor downside. Your economy will suffer a bit during an attack (mostly your production), because your mountain based improvements (Mines mostly) will be pillaged nearly immediately by the enemy and poof there goes your defensive advantage (Typically this isn't too massive of an issue).

• Disclaimer: This is solely from my experience and doesn't apply to every case, I'm just trying to explain the mechanics to help you make better decisions when it comes to defense.

  

Civilization Specifics: Know your strengths and weaknesses.

• Keshik. The Keshik does extremely well at kiting an enemy unit. With its 5 movement and the ability to move after attacking it is a force to be reckoned with. If you're losing these, you're doing something wrong. They're both a great offensive and defensive force. Start by scouting with them, when you find an enemy run into range and attack them and run out the same turn (to avoid getting counter-attacked). Once you've whittled-down the enemy forces using this strategy, switch to offense. Using a same strategy you can move into enemy city-range, attack, and move back out all in the same turn! (Also, unlike other mounted units, they don't receive a penalty when attacking cities!!!)
• Khan. This guy is a super general. He gives the typical combat bonus AND heals your units for 2-15 HP per turn (Vanilla and G&K/BNW respectively).
• Timeline & Research. Your issues have been before turn 100 it sounds. So you probably don't have access to these units quite yet. It's possible but typically you won't (or at least I don't but I never rush Mounted unit research). The good news is Washington doesn't have his either. In fact, Washington doesn't get his unique units until much-much later in the game. You on the other hand only have to wait a little bit longer, survive using the tactics above and below this (@rsanchez1 makes some good points) and you'll soon have your Golden Age (haha get it, word-play hehe).
• Unique Abilities. Neither of your unique abilities (your's or Washinton's) is directly helpful to the situation of war with each other. However, you can levy your bonus strength against City-States to get them in the afraid state. While afraid you can demand tribute (small amount of gold) which can be helpful in upgrading a unit or purchasing another. The best he gets is extra vision, which in the hands of the AI I doubt is very useful at all.
• Summary. Your time to strike is very soon after researching Knights. You should immediately start producing Keshiks and hopefully you'll have a Khan (Great General) by then too. Reduce Washington to nothing (or close to it) at this point. If you allow him to live (comfortably) he may present a problem later in the game since Minutemen are one of the best Unique Units in the game (imo).

Hope this helps, feel free to ask any questions about my tips or things I might have missed that are related to the original question in the comments below and I'll see what I can do about answering them.

You are playing Mongolia. You are a warmonger civ that is more suited to war than America. Plus, you have an incredible mounted range unit in the Keshik. Your strategy should revolve around fielding Keshiks decisively against America.

You mentioned if you don't expand in America's direction, you usually get attacked by turn 100. This will be enough time to get Keshiks, so I would say do not expand in America's direction initially. Before you get to Keshiks, build a few chariots and horses and have them chase barbarians around to get some promotions. These are especially useful for chariots because their ranged promotions translate directly into the Kesik's ranged attack. In this time you will also want to keep your economy going strong so you have enough gold to upgrade your chariots to Keshiks.

Importantly, you will want to keep a few melee horseman around after you get Keshiks to employ Mongolia's advantages most optimally. The Keshik and the Khan will move 5 units. The Horseman will move 5 units from Mongolia's ability. The Keshik is perfect at hit and run attacks. Your strategy will be wear down America's army with Keshiks, keeping them safely out of harm's way. Then move in to their cities, attack with Keshiks and retreat after attacking (Keshiks can move after attacking). When the city is low enough, the horseman you've kept safely in the rear can move in to conquer the city.

Hit and run will be your major advantage against America, and if you can get to America before Minutemen come into play, you will be in position to take a few major cities and seriously set America back for the rest of the game, or even outright conquer them if you have the capability of doing so.