I might try a "lighter" oil - say, a vegitable oil or even jojoba oil - and see if that loosens it some. In theory, the lighter oil will help break up the lansinoh. Then some washes to clean it all away.
(Jojoba oil is very close in texture and pH to your body's own oil, so it makes an excellent hair conditioner. It's very light. And it's a bit pricy. Olive oil is rather "heavy" and somewhat sticky to me if I use it on hair. Safflower oil is sort of in-between the two - pretty light, and less expensive than jojoba and a pH that is very close to skin and hair's natural pH.)
Oh, you might also try washing her hair with dish soap (the kind you use to hand-wash dishes, NOT dishwasher detergent). Dish soap is better at carrying away oils and grease than shampoo (which usually is formulated to leave some oil behind).
Dish soap won't hurt her hair as long as you end the washings with an acidic rinse or product like a conditioner or a 1:9 vinegar and water solution (leave this in her hair).
If you get to a point where her hair is still pretty greasy, but you are done washing it for the day, you might try putting her hair in braids. This will 1) keep her hair out of the way so the greasiness doesn't attract as much dirt and 2) make her hair look nice and neat so it's not as obvious that it's greasy. And then try again another day.
Ann-Marita. I deleted my usual signature due to, oh, wait, if I say why, that might give too much away.