My response:
I'm sorry that you're frustrated, but in your initial post, you asked the open-ended question:
"How would I go about trying to get a settlement from the doctor's insurance company myself ?"
This is the reason I wanted to have a clarification from you, because ". . . trying to get a settlement" involves numerous steps, knowledge of the applicable laws, and experience with matters such as yours.
If you haven't already done so, and in a sincere effort to assist you, you might try going to a college law library and learn all you can about "negligence" and "medical malpractice". Then, from those books, you'll have an idea how to explain your problems in writing, and in terms that an insurance adjuster understands. You'll also be able to make an educated assessment of the value of your claim.
Additionally, you'll be able to keep tabs on the Statute of Limitations, and know when to file a lawsuit if negotiations fail.
Or, if the above is not your "cup of tea", then keep hunting for an attorney who's willing to take your case.
Good luck.
IAAL