Reliable way to downgrade iPad 2 to iOS 6 without SHSH

Recently I upgraded my iPad 2 from iOS 7 to iOS 9 which made it slower and buggy (apps frequently crashing). I have regretted ever upgrading to iOS 7 let alone iOS 9. Recently I found some sites (eg this one) which offer ways to do exactly what I want: get my iPad 2 back to iOS 6.

(I'm not sure what sites / instructions to trust but I do generally trust StackExhange advice. I am concerned what the effect would be downloading software from some of these sites.)

A few parameters:

1. I don't have backups (eg SHSH)
2. I would prefer to keep some data (eg saved games)
3. Idiot proof instructions would be a huge plus
4. I am prepared to jail break as part of the procedure but want to be left with an unbroken iPad

Can anyone tell me a fool-proof and trustworthy way to downgrade?

Edit, why I don't think this is a dupe: I don't think that is a duplicate because there the answer to that general question is that it is no longer possible. But this specific case does appear to be possible in light of recent developments. Plus one answer mentions many devices but not the iPad 2. Finally, that same answer has SHSH as a pre-requisite, my question excludes that.

Apple is currently signing iOS 6.1.3 for the iPad 2 and iPhone 4S. Simply download the appropriate IPSW and Shift+Restore (Option+Restore on MacOS) in iTunes. The signing status of iOS 6.1.3 for these two devices is unlikely to change because they are installed with iOS 5 out of the box. iOS 5 on these devices cannot update over-the-air to any iOS above 6.1.3 due to some technical limitation. So the update path for an out-of-the-box device over-the-air is iOS 5 -> iOS 6.1.3 -> latest iOS.

Note: At the time I answered, only the OTA for 6.1.3 was signed (because the limitation doesn't exist when updating through IPSWs). For some unknown reason, Apple now signs the IPSW for 6.1.3 (perhaps to be consistent). I don't know when this changed and I cannot find any information about it. The whole reason OdysseusOTA exists is because only the OTA used to be signed. Below you can find my original answer according to the circumstances of early 2016.

The lowest version you can go with an iPad 2 and without SHSH blobs is iOS 6.1.3. This is because Apple still signs OTA (over-the-air) iOS firmware for this version. You can use OdysseusOTA to downgrade to this version, but only if you're jailbroken. If you're not, then you can't. And only iOS <= 9.0.2 can be jailbroken currently. Read my answer on this question for more information about OdysseusOTA.

All tools that claim to be able to downgrade are simply forks of Odysseus and/or its process. Odysseus is the only tool that can do such things at this time.

For a video explaining how to use OdysseusOTA, watch this YouTube video from its author.