Would have to look at the law as it existed when your settlement occurred. Presently, a settlement of a minor's claim requires court approval. The court monitors the funds, which can be used for the minor's benefit. The settlement money belongs to the minor, which the parent (who are typically appointed as the guardians of the funds) watch over, keep safe for the minor. You need to see a lawyer about this as there are many facts that need to be looked at to give you am opinion on your case.
You need to consult with an experienced injury attorney for this. Failing to maintain safe equipment is a valid basis for a lawsuit. You are going to want to talk with an attorney who has handled premises liability cases before with success. These can be tough cases, but if developed properly, they can also be very valuable cases.
Medical bills is only one part of the available elements of compensation.
In your question you state, "to the point this office had a lawyer on notice that met with me and is now going to represent me in my accident case." That is concerning!
Administering a double dose is very likely a breach of the standard of care. I handled a fee of these types of cases in the past and in each a breach was found. That said, the issue is going to be damages. There was some harm done (it seems) but nothing permanent - thankfully. I would consult with a personal injury/medical malpractice attorney to go over all the facts of the case.
You should consult a medical malpractice attorney. If an object was left inside of you (that should not have been), then you likely have a claim. That said, the extent of your damages (harm) is what I would want to fully investigate first. Med mal cases are very lengthy and costly. Before heading down the med mal lawsuit road, I would want to make sure that it is financially advisable. You will be able to get all these questions (and more) answered by any good med mal attorney.
The sooner you meet with an attorney, the sooner your claim will move towards a fair resolution. The insurance company for the at fault party has been and will be working to gather evidence to undermine your claim. The length of time the case will take is impossible to accurately predict without a full understanding of the facts (both liability and medical). I would be happy to discuss this with you - feel free to contact me for a free consultation.