In my opinion, I don't think the cozy keet is a bad place. Parakeets in the wild make their nests by hollowing trees, so the cozy keet log emulates this nest. Your keet chose that nesting place for a reason I guess.
However firstly consult with an avian vet as suggested.
Also here in Mexico some we have those cozy keet logs. They are considered to be Parakeet's nests and I have seen couple of people with breeding keets using them as nests so that's why I think that it isn't much of a problem. But you have 5 chicks right? I think a good strategy is just move the hatched babies to the nest box, making sure mom and dad are feeding them there, and leave the eggs in the warm log. It is your choice though, I don't have any experience breeding. So better ask the vet.