For the first season of a new sport, go with the cheapest that you can get of everything, which is what you'll get at a sporting goods chain (like Dick's). $1 mouth guard (they're sized by age range), $30 basic stick (make sure it's for a girl, the starter sticks are usually all the same size), $30 goggles, and cleats can be the kind that you wear for baseball or soccer if you can't find LAX ones. They're very similar. They probably have a stick/goggle package. STX brand has some decent starter stuff.
Also check craigslist or any sports consignment shops. We got all of my son's first year equipment (except the stick) at a sports consignment store. You're lucky she's a girl - the boys have to wear helmets, chest protectors, shoulder pads and elbow pads.
Sizing the sticks are only for older and more experienced players who are going to stick with the sport for more than a season. If she gets serious about it, then she may later want to get a better quality stick with a lighter shaft, custom head, etc. but for the first year or two, the standard cheapie sticks are just fine.