This is just a thought, and may not be feasible, but here goes:

Since I think Hotmail is a web mail application, you should be able to log into it from any computer as long as you have the passwords etc.

Try to find another computer with more speed and faster internet access. Maybe at the local library?

Log into the problem e-mail account. Hopefully it will actually download the problem e-mail. Then delete it. In theory, you should be able to go back to the usual computer and ISP and have it the e-mail work.