After many hours of online research...

... I've decided that going into London is probably too risky with my time limit, and far too stressful. So, I looked around for other options and now I'm getting a room in the "Yotel" at Heathrow. It's a place where, if you have a long layover, you go and you can sleep, shower, watch tv, go online, etc. Essentially, just a mini version of a hotel for Heathrow. It's gotten great reviews, so I think I'll be fine.

I'll be seeing London soon enough, but not on a time limit and not half asleep after a 6 hour flight. Now, when I get to dublin I'll be refreshed and ready to wander the streets!