They used to make splicing tape for cassettes, but I have used regular old Scotch tape. (I will use the term “splicing tape” to refer to sticky tape. The term “tape” will refer to the media tape.
First, you have to straighten out the media tape. You will want the tape wound around its take-up reel as best as you can. Usually, I would use a pencil to wind the tape while holding the tape itself with just a little bit of tension. As you wind it, make sure the tape is underneath the plastic cassette edges, but over the little felt pad.
Wind all the tape except for a couple of inches.
If you have a squeeze clamp, you can anchor the cassette to the table so you can wind with one hand and apply tension with the other. Otherwise, get a friend to hold the cassette.
Once you have the tape straightened and wound on both take-up reels, lay the two ends together. Typically, you are supposed to lay the ends of the tape over each other and then slice them diagonally with a razor blade. Then, separate the two ends carefully until they just barely touch. Apply a tiny piece of splicing tape to the backside of the tape. The backside is the side that touches the felt pad. The front side is where the recording media is.
What I would typically do is place a half-inch strip of splicing tape along the back, and the smallest piece possible on the front, over the splice. Needless to say, you have to cut the splicing tape so that it is narrower than the media tape itself. The easiest way to cut the tape is just cut it width-wise from the end.
Sometimes, I would have to take the cassettes apart in order to rethread the spools. If you can, take apart another old cassette so you can see how it all holds together.
Some cassettes were held together by tiny screws in the corners. Others, I have split with a single-edged razor blade and then transferred the tape to another cassette case that had screws.
I am sure you can find examples on YouTube.