definitely be careful. Even if she has had a spray of water up there she may not like your hand there till she know what your doing. I have to go in from the back between Vida's back legs to do hers. Fras will let you go in from the side. You need to spread the teets apart and clean between them, not the teets themselves. You'll know it when you hit the dirty part.
I think its easier when its kinda dried out. Then it just sort of flakes off. I use my fingers and to get most of it off then my fingernails to kinda scrape the gooey part off. Make sure to have some handywipes nearby (for yourself, not the horse) I find that because its sort of oily/tarrish that its cleans better without water.
Its really not that bad and once Chance figures it out she will enjoy it.
"Until one has loved an animal, part of one's soul remains unawakened..."
- Anatole France