After installing LED headlights, the high beams do not work?

Legally in the US at least installing any light other than what the OEM installed is illegal as they are not approved for use on the street. Plus, as you have found some of the vehicles are set up so that they detect the bulbs are not correct (through the lower current use) and they will shut off the lighting or if the bulbs are a poor design, they can actually damage the vehicles electrical system because they are internally wired wrong.
Now if the OE did offer an LED lighting system you could install the complete system and it should work, but it might also need to have the option turned on in the vehicles programming.