I think it'll make it. Those stems that have leaves, I'd pin them down in contact with the soil to increase your chances. That way, even if the current roots don't make it, you'll likely have new roots from those.

Regarding the water going right through the pot, yes, this does mean that it dried out far too much. However, if this happens with one of your plants, just wait a while (several minutes, I'd say, or even an hour or two), and water again.

Even when the water "pours straight through," a little bit of it stays, and given a bit of time, will soak into the potting mix, which will begin to puff up, and be willing to absorb water again. Just give it a little love, and many neglected things can be brought back.

Mint seems to be a tough plant so I wouldn't worry about losing it. Just keep it watered till it comes back to normal.

I've tossed mint roots on my garden wall, and let them bake in the sun for a day or two before adding them to the compost heap. it rained maybe two days later - and by the end of the week, it had rooted, and sent out shoots.

mints are hard to kill - yours should come back in a week or so.