Ibis (Accor Hotels) is another possibility. Looking at sample dates 10-12 February the Ibis Budget at Whitechapel and the Ibis at Earls Court are within the price bracket. Granted they are not in the Trafalgar Square area but are not that hard to get to on the tube.
Travelodge have a really handy Price Comparison Chart which you can set by date, location and increasing price. I hope this link works- it is set to London for 5 February, increasing price- https://pricefinder.travelodge.co.uk/?checkin=04%2F02%2F2026
One I keep looking at because it is often great value is Feltham. I accept that it is nowhere near Trafalgar Square, but is right beside Feltham Railway Station. That is in Zone 6 (so TfL fares, if on National Rail), a 24/7 bus ride from Heathrow, and is a fast and frequent train ride away from Waterloo.
If price is key that is not a bad location at all.
The biggest drawback to me about Travelodge is that you have to pay extra for WiFi after the first 30 minutes per day. And the breakfast (if you pay for it) is certainly inferior to that at a Premier Inn. Of the three (Travelodge/Premier Inn/Ibis) the Ibis breakfast is best to me personally. And we will all vary on that.
At least any Travelodge room has windows, and is en suite- likewise Ibis.