If I were doing it manually, I’d disassemble the cassette. (It’s really rather easy, but practice on some dead cassettes first.) This would let me inspect/wipe the sides of the reel of tape and contamination inside the case. If necessary, I’d transfer the reel to another disassembled case (like the dead cassettes I practiced on)
The tape is usually held on the take-up reel by a plastic wedge that slides out easily with your finger. Though I’ve spliced many tapes and films in my time, I think removing re-attaching the new leading end to the take-up real is both easier and more durable than a splice. also this lets you discard the “leader” (from the take-up reel to the flap) which is probably as contaminated as the outermost layer of the main reel. The first 2-3 feet of tape don’t contain any video on commercial cassettess, anyway – not even the FBI warning.
You can often find tape “rewinder/cleaners” on sale on the internet or in stores, but the ones I’ve seen are oddly pricey. You might try making one yourself from a $5 battery-powered rewinder by pulling out a loop of of the tape to thread around pegs covered in lens cloth or paper and (the critical step) replacing all but 1-2 of the batteries with a conductor. Cheap tape rewinders often free-wheel at low torque byt high tape velocity with the flap closed; reducing the voltage (plus the small amount of friction you’ve added) can make it slow and gentle. It’s probably better for rewinding, too. I’ve had to fix many tapes that were rewound so fast by cheap rewinders that they pulled the leading end out of the take-up reel.