Instant Search: improve printed output

### Impacted plugin

Jetpack

### Steps to Reproduce

1. Go to a site with Instant Search enabled and start a search with `/?s=`.
2. Try and print the page.

### A clear and concise description of what you expected to happen.

In some situations, it might be desirable to print a set of search of results (like a product search).

If the search overlay is open, I would expected the printout to include the search results on screen.

### What actually happened

The search overlay is completely invisible on the printout in Firefox.

In Chrome, the printout is not nicely formatted.

### Other information

_No response_

### Operating System

_No response_

### OS Version

_No response_

### Browser

Firefox

### Browser Version(s)

_No response_

The issue names no files, tests, or entry points. Reproduce the Instant Search print behavior in Firefox and Chrome, then trace the search overlay's print styling. Done means the visible search results appear in Firefox printouts and the output is appropriately formatted in Chrome.

Written by the indexing model from the issue text.