Instant Search: Appending query string doesn't work for sites served from non-root path names

[jsnmoon]

Steps to reproduce the issue

1. Suppose a WordPress site exists at https://example.com/wp-blog with a Jetpack Search widget enabled. (See the first two test steps in Instant Search: Update prototype with downstream changes #13761 for explicit instructions.)
2. Enter some text into the Jetpack Search input in the above example blog.

What I expected

• The URL should be set to https://example.com/wp-blog?s=search-term-here.

What happened instead

• The URL was set to https://example.com/?s=search-term-here.

Actual examples
You can try searching at this URL: http://gibrown.wpsandbox.me/demo/.

[jsnmoon]

The tricky part of the fix requires us to know what the actual site URL is. If we don't account for this, we'll run into incorrect behavior like so:

---

1. Incorrect approach: always account for the path name when appending a search query string

• Start browsing a WP site at /.
• Navigate to the about page at /about/.
• Type in a search for "hello".
• Expected URL: /?s=hello. Actual URL: /about/?s=hello. Incorrect!

2. Incorrect (and current) approach: always revert path name to / when appending a search query string

• Start browsing a WP site at /wp-blog.
• Type in a search for "hello".
• Expected URL: /wp-blog/?s=hello. Actual URL: /?s=hello. Incorrect!

3. Correct approach: use a base siteUrl whenever we manipulate query strings

• WP site at / + navigate to /about/ + "hello" search => /?s=hello
• WP site at /wp-blog + "hello" search => /wp-blog/?s=hello