When you say it "sounds ok on bottom string," are you are talking about the bottom pickup (the bridge pickup) or "treble"/down position on the 3-way switch? If that sounds okay, and it is only the middle and top positions (middle uses both pickups, top "rhythm" uses just the neck pickup), then it sounds like your problem might be something involving the neck pickup or its wiring. The first thing to do would be to double check all your soldering, and check it against the LP Special II wiring diagram:
If you have an electronic multi-meter you can use it to check the resistance (in Ohms) of the pot you changed. In a LP guitar typically 500k pots are used, but depending on where you buy the pot they can often be off by a little bit, or a lot if they are lower quality, and that could cause tone loss. You could also replace the 3-way switch, as those are fairly inexpensive, and upgrade the output jack if you are comfortable with wiring and soldering. It's not guaranteed that this would fix the problem but if you do it right it can't hurt. You also can check continuity with the meter, and make sure all the ground wires are connected to the right places. As far as the details go of troubleshooting an electric guitar's electronics, check here for a previous answer.
If you can't find any problems with your soldering and your wiring looks correct, I would recommend taking it to a guitar shop to have a qualified tech take a look at it. On the other hand, if you could conclusively determine it was a problem with the neck pickup, you could take this opportunity to upgrade that pickup. For about $50-$100 you can get a new pickup that would probably make your guitar sound a lot better (at least when using rhythm or middle setting), or you could get a pair of pickups and replace both for under $150. Again, don't do that unless you can confirm it is in fact the neck pickup to blame.
In the long run though, if you can't fix it yourself, then either paying for a guitar tech to fix it or buying new pickups might not be the way to go, because either of those options could cost $100 or more. It might make more sense to get a new guitar if you are serious about playing. New pickups in that thing would cost about as much as the whole guitar did in the first place, and for not much more money you could step up to something like an Epiphone Les Paul 100 that I have heard pretty good things about.