I want to combine a jpg with an mp3 and save the resulting file as a 1080p mp4 with 320kbps audio. I don't want to use an online converter, all the "free" software I've installed so far has been a lie and Windows Movie Maker isn't compatible with my OS (Windows 10). Which fully-functioning no watermark freeware can I use to get this job done?
From your description, it would appear that a slideshow program may do as you require. Even a single image could be considered a slideshow. Most slideshow programs allow one to set a duration of each slide and/or match the duration of the audio imported to the show.
A quick Google search for "free slideshow programs" generated many results. One of them is a compilation of 12 available programs, with a list of pros and cons, giving you a better chance to find one you will like.
Alternatively, you could use a free video editor, import a still image, set the duration and add the audio file. Video editors would also allow you to determine the output format, in case the slideshow software won't generate mp4.
I can confirm that the video editor Kdenlive will do all you require, including the no-cost aspect. It is available for Windows and Linux, although it appears the Linux version is less buggy when performing complex operations. Your requirements would not be considered complex and should work fine on Windows.
Depending how confident you are in a terminal, you can't get a better option than FFmpeg. The command you would use would be this:
ffmpeg -loop 1 -r 1 -i image.png -i audio.mp3 -c:a copy -shortest output.mp4
Taken from this YouTube video. It plays your image for 1 second then loops that until your audio ends.
If command line prompts aren't your thing, there are a lot of GUI front-ends to FFmpeg, as well. I haven't used them, so I can't offer any instruction or guidance there, but the FFmpeg core of them is very capable of doing what you want.
Basically any video editing software will do that, my personal favourite is DaVinci Resolve (wait for the video to finish to get the download links). Don't worry, the free (i.e. not "Studio") version does everything you need (and a lot more) and has no water mark or other silly limitations.