Invalid HTML is being generated

## Description

1. Go to https://docsearch.algolia.com/ and open the search box.
2. Here is a screenshot of the elements:

According to the HTML5 spec, a button element cannot be inside an anchor tag. In fact, "there must be no interactive content descendant". Reference: https://html.spec.whatwg.org/multipage/text-level-semantics.html#the-a-element

## Expected Behavior

Valid HTML to be generated.

Start with the search box implementation shown at https://docsearch.algolia.com/ and inspect the rendered anchor and button structure. Compare the generated markup with the HTML5 specification linked in the issue; done means the search box produces valid HTML without a button inside an anchor.

Written by the indexing model from the issue text.