Mares don't need to have their udders cleaned more than about once a year, Heather. Especially if they aren't nursing a foal. Ordinarily, a mare will shed the skin on her udder on a regular basis, and that keeps the area clean. My suggestion to you is to have your vet clean the mare's udder when she's sedated and under restraint (such as distracted by a twitch, for example) for other reasons, such as dental work. She'll quickly figure out that having her udder touched and handled isn't such a bad thing that way, and you'll be much safer.
One note of caution for you, however. I would strongly recommend AGAINST ever attempting to breed this mare, because of the potential for serious HARM to any foal that she has. Not to mention the fact that she will teach her baby that it's okay to kick at people, which is something that NO HORSE ever should learn. Remember the old adage about how it's impossible to raise a good tempered foal from a bad tempered mare? Well, that old saying isn't just a bunch of malarkey. It's TRUE. Most horses only know one parent in their entire lives- their mothers. And for the sake of the foal and its handlers, that mare had better have a good personality, or the result will be a disaster.