Is there a metal corner bead peeking out at the base? You can leave it there, but you'll have a high spot in your mud work there. Scrape off the high spots at the old base, apply self stick mesh tape, and apply a setting type drywall mud like Easy Sand 90 with a 6" knife. Then, after it sets, apply other coats with wider knives.

The new piece of drywall looks like it sticks out proud of the rest of the drywall. Does it, or does it just look like that in the picture.

The new drywall piece also looks to me like it's not flush with the existing wall but that may just be the lighting in the photo. The new piece should be flush with the rest of the wall if it isn't. Before taping the seam and applying mud, take a box cutter and make about a 1/8-1/4inch trim back of the surfaces of the new and old sheetrock along the seam....trim back the paper and a little of the "meat" of the sheetrock (this is easily misinterpreted so look for a how-to on the subject that has a photo!) so you know you don't have any places along the seam where the paper may not be adhered well or perhaps there may be a little bit of the drywall that fractured along the cut. After doing this trimback, prefill the seam with mud and let that dry before applying the tape and mud to finish the seam.

Thanks for the input so far guys. It’s as flush as I could get it. Here’s a top down picture if it helps.

Could it be as simple as filling this joint with spackle or would that crack over time with movement/humidity changes?

Ok thank you! Also for the skim coat, do I need to skim coat the entire nook Sheetrock or just the joints that I mentioned in this post?

You would skim coat and feather out the area far enough that the change is plane is not noticeable.

Makes sense. Thanks

Clean up the seams. Level with high strength compound. Tape (e.g. mesh). Finish compound and feather out. Prime. Paint.

It should be finished by now...