Import external assets from pasted HTML

We're redesigning a website that's extremely low budget. As such, we haven't been retained to re-work any of their content.

Old site = Static HTML
New site = WordPress

We're able to copy/paste the data fairly efficiently into WordPress for each page, but many of the pages have externally linked assets when we paste it in, referencing the old domain/path (as expected).

Is there some magical plugin that we can press a button to auto-import all of the externally linked assets into the Media Library? There has to be some easier way to accomplish this.

I've seen solutions that supposedly work for images, but so far nothing for other types of "files".