Editing low resolution photo blur edges

An 800 x 800 image contains 640,000 pixels.

A 2048 x 2048 image contains 4,194,304 pixels, so about 85% of the pixels in the enlarged image have to be invented.

So this is never going to look good.

I managed to get a reasonably good result with Topaz Gigapixel AI and Topaz Sharpen AI. (you can get a 30 day free trial for both of them) I first enlarged the image to 2048 x 2048 in Gigapixel AI, using the Compressed option, with Suppress noise and Reduce blur set to zero.

Then I sharpened the enlarged image with Sharpen AI, using the Focus option, with Sharpness set to 70 and Noise suppression set to 50.

I just processed the original jpg, and had to save it twice, which is not good for image quality.

The jpg format uses destructive compression every time you save, and if you keep saving a jpg over and over, it will eventually start to disintegrate.

So whichever method you use, I suggest that you save all jpgs as Tiff, and edit the Tiffs.

And here's the original: (800 x 800)

I think involving Lightroom is a waste of time and also image integrity.

You should be able to get the same results with the Camera Raw Filter but I wouldn’t necessarily suggest it. 

But the touch-up (removing the back object) and the fairly extreme color changes make a meaningful comparison between the original image and your resulting image hard in my opinion. 

I think a combination of Preserve Details 2 and Super Zoom (with added sharpening) may suffice while avoiding the artefacts like »double-lines« of your result. 

If you need it to look perfectly convincing, it's extremely difficult and always requires a lot of manual retouching. There's really no way around that. You basically need to rebuild the image.

If you accept that the result will never be as good as an original in higher resolution, you might get off more easily.

First of all, don't apply global sharpening. Do it on a separate layer where you can mask in or out the required parts. General sharpening will always introduce artifacts in places where you don't want it.

I'd recommend using the ACR filter on a smart object layer. Then you can go back and readjust the parameters later. Don't use Lightroom for this, the ACR filter will give the same result and is much closer to home. It's easier to turn into a smart object.

It will look like this in your Layers panel:

Do the same for noise reduction.

In any case, don't upsample any more than you absolutely have to. What is the original pixel size, and what is the minumum target size (and why/what purpose)?