Success rates, and better still recent success rates - a change in embryologist can make a huge difference. The HFEA stuff is way out of date, but is as good a starting place as any. But do phone round a few hospitals to get their latest brochures. The difference in rates can be staggering!
If you're just starting off with IVF and are "plain vanilla" then you have a reasonable selection of clinics. As the process goes forward you may start getting fussier (eg suspected antibody issues, repeated m/cs etc), and you may wish to hone down to one or two specific consultants, but frankly it is a small world and if you current consultant suspects a more complex issue they will often suggest that you go to the most appropriate clinic anyway - must say this took me aback as I assumed that they were all in competition with each other!
You will have a lot of visits so make sure that you can get to the place easily.
You should have an initial consultation ebfore starting down the emds regime etc. Take this opportunity to ask as much as you want. Do check whether you would see the same person throughout or not if that is important to you. I went to the Lister which can be a bit of a conveyor belt (typically 10+ women having egg collection each day), but overall a great team with excellent results (at least they were in 2000 and 2002!).
Price may also be a factor. I think that in the £/baby scale Kings came out near the top, but then again how many attempts do you want overall?