Hi Cheryl,
Can't speak for Copenhagen/Sweden but check out some of the big chain websites for the UK.
Fitness First/Virgin Active/Cannons etc. Then search for a location with roughly the same postcode ie W1H etc. Yes this is the same as a zip code in the US. You are right in the heart of central London so will be able to walk or get the tube once you have a location.
Most gyms have their timetable available on the website too so you can check out which classes you can go to.
Top tip if you can, ring in advance to check if there is a booking policy for classes or if you can just turn up. Also a rough price check would be $20 per visit for a guest so ask if they can do you a deal if you're gonna be there a few days and mention you are an instructor too. When I travel to Singapore I get a discount at the Fitness First I go to although I still have to pay.
My final tip - take a towel and a lock for a locker in case they don't have a key system! I know this sounds wierd but on my travels to gyms around the world (see my user name!) these two things can be a pain if you want to lock your stuff away or want a shower (towels often are chargeable)
Have a fab trip!