Several people on this forum have reported mild successes just by making sure that the space was more-or-less air tight: caulking windows shut, making sure that the door jamb sealed, closing off venting to the space, installing storm windows, etc.
That would be the cheapest place to start a project that is inherently expensive.
Really good advice. Also, you need to look at HVAC ductwork, which can easily sabotage any other efforts, if your ducts in the loud room connect directly to adjacent rooms (usually the case). Sounds like you're renting. Your options are pretty scarce.