Even though there isn't a little flash drive buried in your mini that you could take off to grab those precious pictures, a motherboard itself is just a bunch of little Legos that all do a job. There is no theoretical reason that the logic board can't be repaired--it's just not a DIY thing, and sometimes the time to troubleshoot makes the repair not feasible even when it may be possible.
From my experience, I have two comments
1.) Apple Genae have a limited script for troubleshooting problems--they consider "motherboard problem" a big black box that they don't touch. You've come to the right place to get much more thoughtful advice.
2.) I would bet this iPad could be saved. In the absence of water or a drop, these devices just don't suddenly fail. Have you tried a hard reset--hold down the top power and bottom home button simultaneously fit 10seconds. If no, then I bet your battery is either not charging or not talking to the vbatt line--a technician could open your device and see what happens if the device is hooked directly to a DC power supply.
There are definitely things that can be tried. Feel free to contact me via my profile if you have any questions.
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