Ugh, crappy, pixely quality, and it isn't even the one I originally wanted! The original was still over the limit when I changed it to jpeg. If anyone else has any useful advice, please post.
A proper image manipulation programme will be able to save an image with varying degrees of quality. Most of the time it won’t tell you the file size before saving, so ‘save as’ copies with decreasing quality until you get one that’s below the limits.