Here are some cassette tape repair techniques that are listed in Wikipedia. I have used most of the repair techniques are one time or another.
http://en.wikipedia.org/wiki/Compact_audio_cassette
Repair
Cassettes contain moving parts such as rollers which can become worn, leading to jamming or breakage of the magnetic tape itself. This need not spell the end of a valuable recording: one can extract the tape from the shell of its cartridge and transplant it into a new cartridge. "Eaten" tapes as described above can in many cases also be rescued after running the tapes' back side over a rounded surface such as a pen to re-flatten it as best as possible. A crude repair can even be performed on a severed tape by carefully aligning and reattaching the strand ends against a backing of special mylar splicing tape (normal adhesive tape should never be used on audio recording tape because the adhesive will "bleed" over time), then trimming excess on either side with scissors or a precision knife, enabling a basic playback such as for dubbing out to another recording device.
Most prerecorded compact cassettes are glued or welded, but the vast majority of blank tapes are held together by four or five small screws so that they may be disassembled and reassembled easily. It may be helpful to practice the procedure on the latter first. Care should be taken to ensure that the tape follows the correct path through the cassette before reassembly, and that all of the small parts are in their proper places. Shells that are glued together can be opened by first twisting the shell until a "crack" is heard (or until it becomes clear that it isn't going to give), then inserting a knife blade in the seam between the halves of the cassette shell, working round the edge. Care must be taken not to gouge or damage the reels.
On reassembly, a small amount of polystyrene cement (the kind used in assembling model aircraft kits) can be used along the edges and/or the internal lugs to hold the shell together. If it doesn't appear to be needed, don't bother.
The spools on which the tape is wound are often slotted so that the end of the tape may be freed from one spool and secured to a new spool.
Other common problems with cassette tapes are:
The tape runs unevenly inside the shell, causing speed variations and/or jamming during playback or rewinding. This is often caused by repeated winding/rewinding to selected portions of tape without letting the tape play through an entire side. Another cause is heat damage to the shell. In extreme cases, a loud screeching sound may also be heard coming from the cassette itself and also via the playback head, especially towards the end of the tape. To address such problems, first slap the cassette on a flat surface several times and then rewind the tape all the way through without pause, several times if necessary. (Loosening the screws, or twisting the shell of a welded tape until a crack is heard, may help by giving the tape more room.) If this does not work, open the shell and try replacing the "foils" on each side with some of different design from a donor cassette. If none of this works, try a donor shell - but be aware that even if the shell is replaced, you may find that the tape has become stretched and is unusable. European buyers/owners of second-hand cassettes should be particularly wary of pre-recorded tapes produced by EMI in the 1980's with a ribbed shell.
The felt pressure pad that contacts the magnetic tape eventually comes away or disintegrates. This is less common in more recent cassettes, as the sides of the pad are usually gripped by small protrusions from the copper spring that holds it. The pad can be reattached with a drop of superglue, but if it has been lost, the whole unit can be replaced with another either by (1) opening the shell; or (2) pulling the old unit out through the aperture, flattening the ends of the replacement, feeding it through the aperture one end at a time, and bending the ends back.
Some manufacturers, notably Philips/PolyGram in the 1970s and the early 1980s, used a felt pad attached to the metal shield below by means of a piece of sponge that is prone to crumble with age. This generally can be remedied by opening the shell and replacing the components with a "traditional" setup from a donor cassette.
The leader tape becomes detached from the tape itself—another consequence of age, and generally confined to prerecorded cassettes. Be particularly wary of UK cassettes from the early 1970s in a dark green shell: if buying one of these, or playing one for the first time in several years, avoid, at all costs, winding/rewinding before you have wound the tape through manually with a pen(cil) or manual winder and checked the splice. It's much easier to rectify the problem by pulling the splice through the main aperture and checking/repairing it than it is to break the shell open, especially if the shell is of the "smooth" type with an internal weld that cannot easily be broken without permanently damaging it.
