Accommodation in London is very expensive. Even if you stay out in the suburbs where it's cheaper, it's still expensive. Think I paid140 pounds a night for a pretty ordinary Renaissance at Heathrow. Don't know how much people are letting AirBnBs go for there, but definitely have a look.
Food is reasonable enough, public transportation isn't cheap but not budget busting. Just somewhere to sleep depending on your acceptable standards that's concerning.
Paris is much cheaper in the suburbs than London. Assume your budget for getting to Paris is outside your $100 a day? The Eurostar chunnel train also isn't cheap, could be better off with the likes of Ryanair or EasyJet.