I'd recommend using software that will let you take a "profile" of the noise (by looking at a section of the tape that should be quiet, and so only has noise, no music). Then it uses that profile to try to get rid of the noise in other places. That should do a much better job than just cutting off everything above 10kHz.
CoolEdit will do that, and I think the demo version is fully functional for a certain number of days, so that may be worth a shot. Alternatively, it looks like
Audacity (the free, open-source audio editor) has a noise removal function. I have no idea how it works or if it's useful at all, but I'd be surprised if it didn't sound better than what you're doing now...
