First of all I would love it if a friend sent that to me-wonky or not. That's not what I would see when I looked at it.
Second if you are really bothered by the wonkiness you might try to steam block it. (As long as none of the threads or fabrics in it will be damaged by the heat and steam) On a pressing surface like your ironing board or pressing mat, outline a rectangle of the right size. You could draw it on freezer paper and then lightly press the freezer paper to the ironing board. Steam the mug rug well from just above it (not actually pressing it), then while the mug rug is still damp and warm from the steam, place pins along the perimeter, stretching it a bit where needed to make it match the outline on your freezer paper. Leave it in place with the pins until it dries. Kind of like blocking a sweater. Make sure your pins are of a metal that won't rust. (But again, I don't think you really need to do that-your friend will love it the way it is!!)