How can I change this black and white sketch on a transparent background to all white?

Another option would be a Blend If and a Color Overlay. 

Wow. In this particular case you are home dry, because the line drawing is a PNG with transparent background.

So simply open it in Photoshop

Add a layer beneath and fill with blue.  I also added grain because your example appeared to have some noise.

Set the line drawing mode to Screen.

I have inverted the line drawing because I think it looks better, but it works both ways.

I assume you want something like the image with the blue background. Try this: Command+Click the RGB composite channel to load it as a selection, and invert the selection. With the inverted selection active, make a new layer, and Edit > Fill it with White, and de-select.