Given what you intend to use the new room for, leaving the insulation in the wall should be adequate. Consider the situation, before your new room was there, and the existing wall was just an outside one. It should have done an excellent job of keeping most of the outside noises from entering the house.
At any rate, the use you propose for your new room is not unlike a typical room in the house, and probably much quieter than the typical teenager’s room. The existing insulation should do a pretty good job.
If you are still unsure, the next economical step would be to build a new wall inside the new room, next to, but not actually touching the existing wall separating your work room from your daughter’s room. The idea here is that the sound in your work room will resonate the drywall, then the studs, and finally, the drywall on your daughter’s side. By building a new wall in your work room that is, say ½ inch away from the existing wall, the new studs will not be touching the old ones, and sounds will not easily translate to the other side.
It would also be best to remove all the old exterior sheathing, exposing the insulation, and also insulate this new wall. Then, finish the new wall with drywall and your paneling.