Last minute booking May 10-16 in Edinburgh and need help finding an affordable hotel for 1 person

That is a very low budget for Edinburgh. I actually would be surprised if you can find anywhere at that price point that isn’t a hostel. Possibly a very budget and tired B&B. If I was you, and my budget was fixed, I’d split my trip and have half my nights in more expensive Edinburgh and then half somewhere cheaper.

Very suspicious. Why does it matter what the air fare was?
Why want links, rather than just hotel names, and a near impossible mission.
Isn't asking about an itinerary.

First, check booking.com if you haven't already done so.

I'm traveling at the moment and don't have access to my notes from previous trips, but the place I stayed back in 2017 might be in your price range. I think it was the Destiny Murano property just east of the main road heading out to Leith. I found it on booking.com, but boking directly might be less expensive.

It's called "student housing". The rooms are very small and have what I call "dorm furniture", including single beds, but you have your own bathroom. I also had access to a refrigerator ahared by about 6 rooms.

I walked to the Royal Mile, etc., but it is some distance, so check the mileage on Google Mapa to be sure you're OK with it. There are buses passing nearby.

Edinburgh is not an inexpensive city. I think you'd find Glasgow quite a bit cheaper.

That's £60-£80 for a hotel in Edinburgh... I think you will be struggling to find anything reasonable for that price - and you might not want to stop in any that are that cheap...

I checked Premier Inns. There in-city hotels are out of your price range but some of their suburban hotels are closer to your price range. You just have to see how difficult it would be to go into the city every day. For some, without a car, there is no public transit. You would need a cab and if you're shelling out for a cab, you might as well put that money into a closer hotel.

https://www.premierinn.com/gb/en/home.html

You might also want to look at hotels near the airport. At least from there you can get a tram into the center of the city.

For your budget you could stay in central Glasgow and travel to Edinburgh on the train. Maybe check out Glasgow one or two days.
Can you get a flight to Glasgow instead?

Is this your first trip to Europe? One person traveling, looking for a 'cheap hotel' and doesn't want to stay in a hostel? You may be better staying at home. Check out Rick Steves guide book, even the area around Dalkeith Road may be too much for you. Many may already be booked. May is the beginning of the high season.