Perhaps the best speakers to ever exist in a laptop are in the Dell XPS L702x, which you can definitely get within your budget. Although I realize this post is late, it's worth mentioning for others who happen upon the threat.
Everything else I have ever tried pales in comparison to the L702x. And that includes on the Legion Y720 that I just bought. I was looking for a laptop with modern gaming capability that has speakers like the L702x. The Y720 even has JBL speakers like the L702x and I read reviews that the Y720 has "the best speakers ever in a laptop." Well I got the Y720 today and the speakers are night and day bad compared to the L702x. The subwoofer simply cannot pull the fullness that the one in the L702x can. The Y720 sounds hollow by comparison, even after a good hour of fiddling around with the Dolby Atmos equalizer that's included in the software. If this is "the best" that modern laptops have to offer, I'm kind of depressed. Would love to know if anyone is aware of a laptop that comes anywhere near the L702x but with a modern video card in it, either 9 or 10 series NVIDIA.