What's the best way to patch baseboard trim?
I'm not sure that "patch" piece is actually helping at all. It might very well look better with just some caulk and the large gap.

If you're set on trying to patch it, the first thing is to make sure the patch piece is sized correctly. It might just be the perspective of the picture, but it the piece looks like it might be larger than original baseboard.

After you have the new piece secured (glued) in place, you can use wood filler to fill all the gaps.  Apply generously and try to feather it into the rest of the baseboard. You might need more than a single application.  Once fully dried, sand down the filler so it blends well with the rest of the baseboard, prime and paint!