Hi @galaxybolt ,
Some things to try to start the phone without losing data.
i). Try removing the battery and then holding the power button operated for 15 seconds, then releasing it. Reinsert the battery and try to start the phone
Here’s the ifixit Samsung Galaxy S4 Mini Battery Replacement guide that may help.
ii). Try getting into recovery mode and select Reboot system now or even Wipe cache partition and see if that works.
Here’s a link that shows how to hard reset it but the other options are performed the same way. Just select either of them instead of wipe data/factory reset.
If you do get it to turn on, backup the phone to a computer or the cloud, using the phone’s backup feature. That way you will always have a copy of your data if it fails and the only option is to hard reset it. You will have to do this regularly though because your data will be changing as you add and delete things from the phone.