You're fine posting here, although you might get more responses on the German forum.
When I was in Berlin in 2022, I stayed in Hackescher Markt, at the Adina Apartment Hotel Hackescher Markt. My room was a studio since I was by myself but they also have 1-2 bedroom apartments with fully equipped kitchens, washers and dryers and so on. They do have hotels in other Berlin locations as well, but I personally really liked this location. The staff was friendly and courteous, and the rooms were very well prepared, large, clean and nice. Breakfast is available although given the kitchen facilities, I just ate in my room in the morning. And the price was quite reasonable and I would think within your budget.
The hotel is very convenient to the U-Bahn, S-Bahn, and trams, and is an easy walk to the major museums. The DDR museum is only a few blocks away, along with the TV Tower, and Museum Island is a quick 5 minute walk. You can easily walk to Lindenstraß from there, There are several grocery stores a few blocks away along with a couple of wine shops, many cafes and good restaurants. I would definitely stay in this location again.