Though you can certainly get a set of self-powered speakers in that price range, I doubt that they will come with a remote. Which means that you'll have to manually adjust the audio continuously. Worse yet, the sound quality may not be that much better?
I'd invest in a soundbar, assuming that you can figure a convenient way to mount it. Most all come with wireless remotes.
In fact, I just picked up one from Walmart, an Energy soundbar made by Klipsch. Walmart sold it originally for $184, then discounted it to $131., and now have it on clearance for $98. It also has Bluetooth, so can play back music from tablets or smartphones. I connect mine using an optical cable(not included) from the HDTV to the soundbar.
It sounds fantastic. My test was playing 2001: A Space Odyssey. Beats my LCD HDTV built in speakers hands down. In fact, could never fully make out clearly what Tesla(David Bowie) was saying in the movie The Prestige. The mystery finally solved. :B