London hotels with kids really mean Travelodge, Premier Inn (a new one near Liverpool St giving good deals recently), Novotel, the All Seasons in Southwark, or gambing on a limited number of independent hotels. London hotel rooms are small and expensive.
There is a hotel opposite the Albert museum we were going to book before but changed plans , it has a swimming pool and is family friendly , unfortunately I've forgotten the name ,but you might get it if you Google it. or use Google earth , we were traveling 2+2 and price was good , reviews seemed fine too , good luck !
I forgot to mention Holiday Inn & Express HI in my list of London family chain hotels above. Perhaps there may be others, or recommended independent hotels.