I'm a liege without an heir. My wife is barren. What do I do?

I'm a 28yo king of Eire with an old wife who can no longer bear children. (Don't ask me how I got here, things happen in Crusader Kings 2.)

I've tried to get rid of my wife, but the Pope won't give me a divorce. Neither can I find backers to kill her (I suppose the fact that she is a duchess somewhere in central France adds to that).

One option is just to let her die of natural causes (she's 20 years older than me), since I could remarry after and have a new young wife bear children. This will likely incur some period of regency, but oh well.

The other one is to get some lovers, sire bastard sons, and legitimize them. Worse than natural heirs, but not a game over. In this case, how do I increase likelihood of getting lovers?

When I die, the game is over and I don't want that to happen. Hence, my question is, how do I get an heir from here? Which of above options should I follow and is there a better way?

Divorcing her

So, the pope won't let you divorce her? Start your own papacy!

Set your investiture to free investiture and find a suitable bishop. Preferably one that likes you. Give him gifts, honorary titles and make him Court Chaplain if necessary- you want a content yes-man here. Make him anti-pope.

As an anti-pope, he will be able to grant you the divorce you want.

Killing her

Having her assassinated is apparently not an option. Fortunately, you don't need to.

Arrest her. Execute her.

I recommend this only as a last resort. You'll earn yourself a whopping -50 malus from all your vassals for tyrannically arresting and executing someone for no reason.

Also, this might not be possible, if she's a duchess outside your realm and not actually your vassal.

In addition to what other people have already said about ways of getting children, there is also another option. If you have relatives of your own blood you can try to institute elective monarchy (arguably the best succession system anyway) and put your chosen relative successor in charge of a duchy (it's also worth checking out their children/grandchildren especially if you're doing this at the age of 28).

Also, the legitimized bastard trait is not so bad.