You should not accept yourself as a failure and you need an effective sleep medicine.

If your supervisors seem to think that you are doing a good job, I would go with their opinion and judgment. They are probably more experienced than you are in assessing the reports' performance. I would not worry and engage in second-guessing. That said, if you go on with insomnia for a long time, your performance WILL suffer.

A GP should be able to prescribe for insomnia - you do not need a psychiatrist. A psychiatrist is better but not necessary.

At the least while you are figuring it out, use Benadryl. It will be effective for a short period of time while you are getting a prescription.

Good luck!