can i sue the person who hurt my arm

You can sue the person individually, but we would almost always also include the hospital/employer in the lawsuit.  It is no more work to include the employer, and doing this helps to ensure that there will be insurance or assets to pay a verdict.  The bigger problem that you face is that even if you sued only the individual, you would still need to file this as a malpractice case for the failure to take appropriate care when drawing blood.  In these types of cases, a person needs to have catastrophic damages before we can take his or her case.  They are very costly to bring.  This is probably the reason that the 7 lawyers you consulted with all rejected your case.