Budget Hotel Recommendations?

Premier Inn is deservedly the most liked budget chain on this forum, with many properties close to London tube stations, and should have rooms within your budget.

Travelodge and Ibis (Accor Hotels) are two other safe budget chains with multiple central London properties to look at. The price point of Travelodge is often below that of Premier Inn, but they have unbundled like many airlines, so be careful to make direct comparisons between PI and Travelodge. Travelodge rooms are more budget than PI, although quite acceptable.

When is the trip? Soon?

Check to see The Resident Kensington website:

https://www.residenthotels.com/kensington/

Hi, Gina, I recently booked a room for 5 nights at the Premier Inn County Hall (in South Bank) for March of this year, and couldn't be more pleased with the location. I've stayed at Premier Inns in the past and they are very consistently good. I would never consider them a "charming" hotel but you will have no unpleasant surprises there. The rooms are clean with very comfortable beds and with breakfast available (not included but reasonably priced). Rooms have coffee and tea makings with other hotel amenities available and the prices are great. For a standard room at the PI I am staying at, their prices were around £120, but you might be able to find rooms even cheaper at other locations. They have multiple locations in London and I recommend them for anyone on a budget. https://www.premierinn.com/gb/en/home.html

Here are a few threads that will give more info:

https://community.ricksteves.com/travel-forum/england-reviews/premier-inn-f415b5a6-ac53-4fb6-83a3-34f14ae6c4db
https://community.ricksteves.com/travel-forum/england/hotel-recommendation-for-solo-woman-in-london
https://community.ricksteves.com/travel-forum/england/requesting-opinions-of-certain-premier-inns-in-london

$175 is about 140 GBP. That is very low for London.

When is the trip?

Premie Inn is a great budget hotel. I’ve stayed at the one in Ealing before and it was cheap however a little far out of city center. This trip I decided to stay at the Hammersmith Sheperd’s Bush location. Price was very reasonable and it’s a 2 minute walk to the tube station.

Some Premier Inns will hold your luggage before check in or after check out, whereas a Travelodge I checked did not have that service. That's really important if you have a full day of sight-seeing after you check out.

Budget? Not sure I'd call $USD175 budget.
If you are looking for serious budget accommodation try Wombats.