I use a cassette splicing block and ordinary splicing tape (for 1/4 inch, since that's what I have around), but I do an improvised method. Splicing tape is always applied to the inside, which does not touch the heads.
Assuming you open up the shell and access the spools:
-de-wrinkle the broken tape ends and match them up in the block. If they are badly damaged, you may have to cut off a small amount of tape: overlap the ends in the block and use a single edge razor in the cutting groove to make matching ends.
-I use ordinary splicing tape, but cut off a small piece and place it cross-wise over the splice with excess hanging off the sides. Then I remove the cassette tape with the splicing tape hanging off the sides, and place it on a piece of paper, temporarily rubbing the splicing tape down to hold it. Then I use the razor to trim the splicing tape off of the sides. This is a bit tricky. I use my magnifying visor and try to carefully place the razor directly on top at the edge of the cassette tape, trying to match up the side of the tape and the razor, then just push down to cut. IMO, a teeny concave cut into the tape sides is preferable to excess splicing tape hanging off the sides, but best to try to make it perfectly straight.
I have to use tweezers for much of this process. I suppose it is possible to do this without a splicing block, but the block really helps keep things straight while splicing tape is applied.
If the tape is wrinkled at all, it will never play cleanly again, sorry.