Canary Wharf has relatively high council tax. I was actually outpriced out of that area 5 years ago, it became *much* cheaper to pay mortgage on house in Kent than rent 2 bedroom flat round Westferry Rd and pay all the bills, fight for parking space every day etc etc.
Anyway - you'll easily find plenty of places to rent for less than £200 in the area - Mudchute is actually pretty great, there are literally tons of flats in Telegraph Place area, it was one massive buy to let area back in the day. The four towers in The Quarterdeck are excellent for rental - look very seventies, but fantastically quiet, sound insulated, very secure, prices are cheap and if you pick 4th floor and above you'll get river or Canary Wharf views. Going further north, Poplar/All Saints/Westferry triangle is cheap and is getting better and better. Avoid Balfron Towers in All Saints - loads of flats to rent cheap (even split level appartaments), but it's a grim, grim place, and it's one of the very few 1970ies estates to have grade listing, so no one can even paint, fix or renovate stuff properly because it's "historical or cultural significance".