I thought I would tell you of my experience because I had the exact same problem as you.
I had had my iud in for 5 years. I went to my doctor to get my iud removed and she broke one of the strings. This was very painful and she stopped there and referred me for an ultrasound scan. The scan confirmed that the iud was embedded in the womb but not perforated. I was then referred to the womans health unit at the hospital. At my consultation the surgeon had another go and broke the other string. Very painful and I was cramping for a while after. Since there were no strings left, there was no choice but to go for an operation under general anaesthetic.
It was about 2 months until the operation, and I fretted and worried about what I had done to myself. Especially if I couldn't have anymore children, because I desperately wanted more.
Anyway, I had my operation yesterday, and absolutely no dramas at all. I cannot tell you how relieved I am. It took about 30 mins and I went home not long after. The general anaesthetic though knocked me for the rest of the day, but that was it.
Fertility is unscathed, they just said let the womb heal for a month or two before trying to get pregnant.
I'm not sure about your particular case, but it sounds a lot like mine. So, I think everything will be alright. One thing is for sure though, I won't be getting another IUD!