Manually set language

Is it possible to manually set the language for a page that is NOT a wordpress template.

We are pulling in data from an API and using rewrites to display data dynamically.

I have 2 languages

Welsh and English.

If you follow the link in the the box you can click into a single post. I want to then be able to switch between welsh and english.

The dynamic pages are defaulting to English and pulling in the English menu.
I have now create a welsh header and an english header (which is fine)
I want to tell the Welsh header to use the welsh menu.

Or just set the default language for that page to Welsh.

It sounds an easy thing to do? Is it possible?

Thanks

The page I need help with: [log in to see the link]