“Is my hymen broken?”

Your hymen is usually not a complete membrane across your vagina (if it is, you will have other problems, since your periods would not be able to come out). So, whether it is "broken" or not shouldn't make any difference for artificial insemination. The hymen is stretched or tears with sex because it is smaller than the penis. Usually, artificial insemination instruments are smaller than that. The only thing that might make you uncomfortable would be the speculum - the thing we use to see your cervix. We have smaller ones for smaller people, so even that shouldn't be too uncomfortable. At your age, a small syringe to deposit the sperm inside the vagina should be good enough, since it doesn't have to be put into the cervix directly, in which case, you don't need to use the speculum at all.

If you are having regular intercourse it is likely broken. The best way to tell is by your Gyn doctor by exam. You would need an exam to determine that.