You will probably get as many recommendations as responses, but it will likely come down to what's available to you locally.
In general, in your price range, a vintage low power receiver and a pair of 8" or 10" 2-ways should fit within your budget. Certain brands (Marantz, especially) tend to fetch premium prices, but there are LOTS of options. The good news is low power, or entry level models, are both more common on the used market and not as collectible as the TOTL models. Plus, I just like the simple, clean look that comes with fewer knobs and buttons. Since you are in a small dorm room, you don't need a ton of power. You'd be surprised how good 10 - 20 watts can sound with a pair of moderately efficient speakers in a small. confined space.
I'm currently running a little Sherwood S-7100A (14 Watts Per Channel) in my dining room dedicated vinyl system:
I just love the look and the sound of this little gem:
For speakers, a good place to start would be this thread:
Your Favorite 8 inch 2-Way Speakers
The speakers shown above are a pair of JBL 4301B Control Monitors. They are a great sounding vintage 8" 2-way. Depending on your location, they may be difficult to find in your price range. If so, try looking for their fraternal twin JBL L19 or L19A. They are the consumer version of the 4301B Control Monitors (same drivers, similar sound). They tend to be more common and fetch lower prices than the 4301B:
Most JBLs of this vintage will need the foam surrounds replaced. This can be a negotiating point for getting a good price. A set of replacement foams can be had for about 20 bucks, and if you're reasonably handy and patient, a good DIY project that can be completed in a single evening.
There are LOTS of other options in 8" (or 10") 2-ways from other highly respected vendors (EPI, ADS, Boston Acoustics, etc.). It really comes down to availability, price, condition and personal preference.
Whatever speakers you get, get them up on a pair of stands to get the tweeters at ear level and to clear the sofa on the left side of your diagram.
System synergy is also a factor. Some find the JBL's a bit forward, but I have to say, you could do a whole lot worse than a pair of vintage JBLs driven by a Sherwood S-7100A and a vintage Shure cartridge on your turntable. I can personally vouch for the synergy of these bits when used together. The little Sherwood has a warm, lush tube-like sound that nicely tames the JBLs. Vintage Shure cartridges are plentiful, and don't be put off by Shure leaving the business. There are LOTS of aftermarket replacement stylus options.
And, since you're into vintage, a Shure cartridge, a Sherwood receiver and a pair of JBL speakers would have been common in dorm rooms of the 70s and early 80s (again, I speak from experience).
. I have included a rough picture of how the room looks below.
