Lightroom Classic can darken a photo without the need to add pixels. It can also do some sharpening without adding pixels.
But no... Lightroom can only add pixels when you export, so essentially you do what editing you have with the pixels you have and then if desired increase the number of pixels (but really, I doubt this will help making the image darker or sharper).
Why do you think adding pixels can help with these tasks?
Adding pixels to a digital image will just make the image bigger in pixel dimensions, so the answer is no.
To solve the issues you describe about your image, try the contrast slider and the black slider in the basic panel. If you move the black slider to the left, the darker areas of the photo will get darker.
You can also try the curves adjustment panel. There, you can click on the very bottom left of the curve and move it to the right to get the darker areas to go darker.
I belive that Sally is not using the correct terms here. I belive that she simply wants more contrast in this image.
If so, what she needs to do is to open the image in the Develop section and push the "Contrast" slider over to the right.
As far as sharpening, she can go down to the Detail section in Develop and the top controls will do "some" sharpening. I emphasize "some" because a blurry photo is a blurry photo and software can only do so much.
But I have a strong hunch that a photo that was taken 100 years ago was not initially digitally created. I STRONGLY suggest you consider rescanning the image but do the same adjustments IN THE SCANNER software to enhance the image. You will get much better results if you do these basic initial adjustments when capturing the image than afterwords in any photo-enhancing software.
I strongly encourage you to google "increasing contrast in an image in Lightroom" and "increasing contrast in an image when scanning" for greater details than I can supply here in this forum.
Let me add one small example: here is a photo I took of a local mountain from a great distance with not great atmospheric conditions. Please look at the histogram on the top right. Be aware that the further the "hills" of the histogram are further from the edges, the less dynamic range the image has (less contrast).
One of the first things I did was to bring the "Black" slider to the left to make the blacks "blacker" and the white to the right to make the whites "whiter." All adjustments were made by eye moving this back and forth until I saw what I liked. I would have spent more time on this but didn't want to take the time. I also pushed the Dehaze over to the right a bit as well as the Shadows. Interestingly enough I pushed the Contrast down some. Please note that there's still a whole lot wrong with this image, just showing you the possibilities. And I repeat, you'll get much better results if you start adjusting at the scanner.