What's the best way to soundproof a room on budget

If you add drywall also consider using Green Glue between the layers. It is not a glue, but rather a sound deadening compound. You can also use mass loaded vinyl.

This website has some good information on various methods and their effectiveness.

Sound Proofing Company

The biggest problem is deep bass. It is almost impossible to stop without major work.

Agreed - if you can do an isolated second wall that is a great solution. That option is discussed on the website I gave above. It is a great solution, although not always the easiest and less expensive.

You should also look at the resilient channel solutions, although they can be somewhat complicated to install.

Nsgach - sorry, just providing ideas. If you do it your self, you can do these projects pretty cheaply. But I do agree that double walls and the like can be expensive if you have to pay someone to do it. In fact, I said that a double wall was not inexpensive. But also, this is not a very big wall.

My first recommendation was dry wall with green glue. Did you read the website I referenced? One level of drywall gives a STC of 40, adding a second level goes to 42, adding a second level with Green Glue goes to 52. Using Green Glue with drywall will do a lot more than just adding drywall. Green Glue is not cheap, but it really is effective. Mass is great, but isolating the surfaces is very efficient. Green Glue does a good job of that, by essentially turning the sound into heat, as I understand it. If you have not used it, you should look into it.

1 Layer of Drywall STC 40
2 Layers of Drywall STC 42
2 Layers of Drywall with Green Glue STC 52

They recommend 2 tubes per 4x8 sheet. Green Glue has their own website with recommendations and pricing. 12 tubes is about $150. It is not cheap, but works better than just using straight dry wall. I used in the ceiling of basement when I finished it for a theater. Really does cut down the sound transmission. There used to be a lot of discussion of sound proofing on AVSform in the Home Theater forum. Not sure how active it is these days, but still lots of good information. Ted White from my previous link used to be active there. That is how I got to his site.