My companion won't wear her armor

She won't use Elven stuff because it's weaker than her heavy type Steel armor, I guess because she's skilled in Heavy Armor and not in Light Armor. I've given her Dwarven stuff and she equipped it on the spot myself.

As for why she did equip the elven helm, well she doesn't have a helm at all in her default gear so any helm is better than what she has no matter the stats.

Lydia has a set of default armor that you cannot see in her inventory (unless she's dead). She'll only use stuff that is better than what she currently has. Try giving her something better than steel (higher armor rating). Then you'll see her put it on.

Have you tried dying after giving her the armor? I was in a Dwemer ruin with Illia, a mage I'd picked up near Riften, and as soon as we left the deepest part of the ruin she took off her Mages robes in between load screens and ran around like an exhibitionist, though she had equipped the necklace and ring I'd given her earlier. I couldn't do anything to get her to put her clothes back on until I died in the next ruin we came upon. The autosave kicks in at the ruin entrance and bam, suddenly she was wearing the new robes.