So, slf4j is a facade, and it looks like Solr uses JDK logging behind that. I'm assuming this is correct.
It seems that the best way to do this is to replace the LogLevelSelection servlet with a Handler.
Coding a handler to display current settings is easy, and I've already done it. However, to code the update side, requires a decision on suitable request parameters.
The biggest question is whether to allow multiple settings to be changed in one request. The current LogLevelSelection servlet allows you to change them all in one single hit. However, allowing this in a new Handler would, in my opinion, give rise to an ugly API. Therefore, I suggest that:
http://localhost:8983/solr/admin/logging <- would display current settings
http://localhost:8983/solr/admin/logging?category=core&level=FINE <- would change a single value
This latter would probably output the same as the previous, but perhaps with an 'updated' attribute set to true on that category.
Given the ajax nature of the new UI, I suspect this would be enough for it to work with.