How can I repair a bed with adjustable head (manual) that is broken?
That is called channel steel or U-channel:



Channel steel is available in a wide range of thicknesses and sizes and available in different grades of steel and aluminum. It's going to be hard to find an exact match for that piece on eBay and if you get a stock piece that you have to cut and drill yourself, you may not have the right equipment on-hand to do so.

It you can take the piece off and take it to a local metal fabrication shop, they may be able to make you a replacement piece - they may not have an exact match in stock, but they may be able to fold you a custom piece. They may also be able to make a stronger piece or reinforce it where it bent. Or may even be able to straighten out your piece and spot weld reinforcements to keep it from bending again.