I'm considering adding an MIT licensed library to my code as a raw C file. Currently, my library is only licensed under my LGPL license. If I was to add this MIT licensed C file and its corresponding header to my project, how would I implement the license, and would my users also need to add the license to their own code?
I will assume that the LGPL library in your second sentence refers to your own code and that you plan to distribute the MIT library with your library. You will have to include somewhere in all your distributions (most likely in a LICENSE file) a notice like:
License for MIT Library ======================= <paste MIT library's original license here>
If anyone decides to use your library and include the MIT library's content in a distribution, they will probably* have to include that notice in the distribution as well.
None of this restricts how you can license any of your LGPL library.
*There is/was a small amount of ambiguity whether binary distributions of MIT software need to include a notice.