[BUG] Mobile header layout bug

**Does the issue belong in this repository?**

Yes

**Describe the bug**

On mobile devices, the header lacks sufficient horizontal space to display all its contents. As a result, part of the search bar and the hamburger menu in the top-right corner are pushed off-screen.

In addition, the light-dark theme toggle is also at a weird spot that does not align with the underlying toolbar.

**To Reproduce**
Either open [docs.vulkan.org](https://docs.vulkan.org) on a device with a small screen, or open it via the browser's responsive design mode

**Expected behaviour**
The website has a mobile-friendly layout where all header elements are visible and properly aligned

**Screenshots**


**Smartphone:**
- Device: Samsung Galaxy S24
- OS: Android
- Browser: Firefox for Android, Chrome
- Version [e.g. 22]