The first suggestion would be to give it a good cleaning. You've done that. There are some other area's that can be easily troubleshooted if the cleaning does not work:
1. Check the extractor notches on each side of the chamber. They get clogged with soot and debris and can prevent the bolt from going forward enough for the firing pin to make contact with the cartridge. Often this area looks clean, but is not. A set of dental picks works real well and can be found an the local hardware store or Harbor Freight. Tooth pics can work in a pinch with some solvent, but are not the best.
2. The firing pin needs to be able to move freely back and forth in its slot. With the bolt removed, you should be able to rock it back and forth and see the pin slide to and fro.
3. The firing pin may be worn. It can appear to be normal, but be worn down ever so slightly as to prevent a sufficient strike against the cartridge. I experienced this on a Mod 60. The bad pin was .010 shorter than the new one and caused inconsistent strikes. A replacement from Brownells quickly remedied the problem. Even with the bad pin it would still fire, but inconsistently.
Once cleaned and inspected you may also have to experiment with a few types of ammo. Some guns seem to work well with different brands. For me, just about anything Federal produces seems to work well across the board.
Usually with the Mod 60's all that is required is a good cleaning. They are neat little guns and have been produced in such a variety of styles that it can become addicting to own them.
Good Luck and let us know how it works out.