Hi C. M,
I second the use of goggles. Make sure you get good ones that can be adjusted fully to her face. The trick to using most goggle is to get them wet first, then push them to around the eye sockets so they don't leak.
My son was a born little fishy, but my daughter was a bit more hesitant. I started teaching both of them to swim when they were young by playing lots of pool games with them. Take your daughter to a public pool -- the cost should be pretty low, or to the Y as others have suggested and have her dive for rings and pool torpedos in the shallow area (wearing the goggles, of course!). This will help her get used to being under water--with her eyes open--in a really fun way, and since you're near by, it should help her feel more secure. Another thing I did with my kids was get a hoola hoop and had them swim through the hoop at different levels through the water: at the surface, just below the surface, and than deeper down. As their skills grew, I would move further away from them, so they had to swim greater distances to get through the hoop.
Since your daughter is 10, have you tried asking her why she's having a hard time? She may not have a fear issue, but might not like the feel of water clogging her ears. Using a product like SwimEar or AuroDry will help to unplug her ears quickly after swimming. If she knows she won't have to walk around half-deaf for several hours after swimming, it might help.
Two other thoughts:
1. Teach her the back stroke, or the rescue back stroke, or even the breast stroke. All of these are legitimate swim strokes, and don't require that the face be in the water.
2. Have her stand in a comfortable level of water, put her face in the water, and blow bubbles while you count to 5 (you can do a count down under water using your fingers). The swim teacher should have started with teaching the kids to blow bubbles while swimming, because you can't take another breath until you've blown out the one you're holding. Once she gets the hang of breathing out by blowing bubbles, have her hang onto the side of the pool while she tummy floats and blow bubbles, then practice turning her head to the side to breath. Once she gets that skill down, have her do the same by swimming around the pool hanging onto a kick board -- see the progression? The final step is, of course, adding in the side breath while stroking. If you stroke properly, you should actually make a bit of a trough in the water near your face so that you only have to turn your head a little bit to catch a breath.
Good luck!