I realize that you are new to the job, and that major bills often go to the big dogs. But this is an opportunity for you.
You were elected on a wave of smaller and less government. Privatization is a big part of this movement. You can become a national hero in one fell swoop, with conservatives and many on the left even. Introduce a bill to privatize congressional health insurance. Who would dare oppose you? OK, you might have to protect yourself from bodily harm from your fellow legislators, but isn't that what it's all about, integrity in office? Go to it man, get in there and fight for your principles.
I'm sorry that this is so short, but good ideas are often simple and in need of little elaboration.