Both London and Paris are extremely difficult asks.
I have scored a Getaway in Paris that was an excellent value. But it was Paris for Christmas -- not high summer season.
As long as you're reasonably fit, any place near a Metro station is fine. Doesn't matter which arrondissement you pick. Last time I was there, I was all the way out in the 19th. No big deal. It was GREAT staying far away from the tourist crush. And then taking the Metro to whatever we wanted to see that day.
The shopkeepers around the 19th were particularly friendly, especially considering how badly I mangle the French language.
If staying near "the sights" is a big deal (and I don't think it should be), I'd go with something in the 7th (spendy), 8th (more spendy) or 15th (much less spendy).
I'm easy in Paris -- just point me in the direction of Les Halles, the Bouillon Chartier restaurant or Rue Cler. And then get me to Epernay because I much prefer time in wine country.
As for London, I spend a day or two there and then get out -- anyplace easier on my wallet. Don't get me wrong, London is one of the great cities. But you cannot possibly see it all in one trip, or twenty. Just assume you're going to return, pick a few things to see, and go find a smaller city (I like Salisbury and Glastonbury).