Well, I've tried something new. I actually did it a little differently on both sides because I was experimenting.
On one side, I took a piece of plastic cut from a shopping bag and folded it over the holes. Most of the taping is done from there onto the outside, with no tape inside, past where the lid fits. There's a little bit of tape on the inside, though, to hold things down. At least this time, it's not against the glass--it's solidly on the plastic rather than stretching over more than one surface (like a cord on glass). On the other side, I plugged the holes with bits of paper towels. For the cord, I wrapped the part of it that goes through the hole in paper towel so it fills that gap completely. The towel's held with tape, but I think that portion's outside, and either way, it's all the way around the paper towel. Anyway, I put the lid on, and I filled up any additional tiny gaps with either plastic bag or paper towel, and then taped those down on the outside. So...it's complex.
There's another piece of duct tape, though, over a gap that formed in the mesh lid. In that case, I don't know what to do about it exactly--well, besides buy a new lid, yes, but I suspect it'd happen again. I don't recall it taking any real damage to cause the current gap.