What's the best way to fix heavily stained white Ikea furniture?
Isopropyl alcohol, "rubbing alcohol", will remove most "permanent" markers (e.g. Sharpie brand markers), but may be ineffective on ballpoint pen marks, depending on the ink formulation. For this purpose, don't bother using anything less pure than the commonly available 91%, with 99% being preferred. At 70% concentration it's commonly not effective.

For marking which was recently applied, the marks will usually be removed quickly and completely. Some surfaces, particularly if the writing has been present for an extended time or if the surface is not smooth, may need considerable rubbing or scrubbing (e.g. with a toothbrush, to get some agitation in nooks and crannies), and/or may need to have the isopropyl alcohol sit on the surface for a time alcohol (e.g. with a paper towel dampened by isopropyl alcohol).

There are also stronger removers available from retail. The product "Goof Off" is particularly effective to remove a wide variety of substances, including ink, but will damage some surfaces (e.g. it will partially melt some plastics). If you consider using it, you should test it on an unobtrusive location to see if it causes damage.

Wall paint seems like a terrible idea, will not be as smooth, and will get marked up again. I'd tend towards being happy that your daughter is making art and not worry so much about things being pristine and unmarked.