Three year olds just seem to need a lot of space and wiggle room in the bed. They kick, they move, they turn sideways (and woe to the person who is on the foot side). They simply don't seem to like being cramped in the bed. (I have a 7 year old and a 4 1/2 year old who did the same thing.)
My advice? She needs to sleep in her own bed. It may be a difficult transition for you both, but if you're not getting sleep, it's not good for you or your family. And medicating her for a normal behavior is not good either. Motrin is hard on the digestive system, so use it with care. (Same for Tylenol and the liver.)
I love snuggling with my kids. They slept with us as little bitty ones. When they reached the kicking stage, out they went. Now we find other ways to snuggle. We watch movies and eat popcorn; we read stories in bed every night; we sit and talk about things; I snuggle in their beds at bedtime and listen them talk about their day. And in our bed? Well, I'm a SAHM and when my husband gets up for work, the littlest one usually hears him and comes in our bed to snuggle with me for a while (and the oldest did the same when he was about 4).
Think up some snuggly ideas of your own -- you work full-time, so maybe a Saturday morning snuggle time, a Friday movie night curled up on the couch, extra time snuggling at bedtime to hear about her day. You can do it!
It's hard to let them grow up and move to the next stage. We want the snuggly little baby (without the diapers and the crying). We have to let them have their own space. This is only the first of many ways in which you will have to give your child her own space.