I don't think that we actually need to spend extra effort to have Eike fight something. It's a dangerous world and with mage armor she is better able to handle it then most people.
Just send her on tasks that are actually useful and use the extra time saved to tutor her.
I think I disagree for two reasons
Firstly that's an assumption that I don't think is entirely safe to make
After all Mathilde managed to make it through the entirety of her Apprenticeship without ever learning to fight properly at all, so clearly you can't rely on everyday experience alone to be your teacher
Secondly that's kinda just asking for things to go wrong at critical moments
Like I said before, part of the point here is to learn Eike's limits in a relatively controlled environment, where if something goes wrong we can be nearby and ready to intervene
And so we can learn and properly gauge what Eike is and isn't prepared for, as well as what we do or don't need to worry about teaching her
If we just bring Eike along into dangerous situations then we're effectively just throwing her into the deep end and trusting in her ability to swim without actually knowing if she can
We know she has combat skill and knowledge, but we don't know how she'll actually react in a combat situation
If we drag her into a dangerous situation what happens if she freezes up in actual combat? Or what if she freezes up
after killing someone? (Again, Mathilde was heavily affected by her first human kill and took some time to collect herself)
Taking an action to actually test Eike in a combat situation also means she's more prepared for when it's unexpected. If she familiarizes herself with situations where she's expecting danger then she'll have a better idea and instinct for what to do when danger sneaks up on her. As opposed to just letting her first experience be danger that she wasn't expecting or prepared for
If it's a dangerous world, then surely it behooves us to adequately prepare Eike for those dangers
