NOTE: Batteries are removable, but some are much harder to access than others. If this is the case with yours, and especially if it is still under warranty, then you should contact the manufacturer about a repair or replacement rather than following these steps.
Try the following and see if it will turn on...
1. First make sure the device has been charged.
2. Unplug your charger cord.
3. Remove the battery.
4. Press and hold the "Power" button for 30 seconds and see if the device will turn on.
5. If it does, great, if not yet, no big deal. Now plug back in the power/charger cord. (If it didn't turn on without the cord in, it should turn on once you it is in. Yes sometimes they will turn on, or try to turn on, even with nothing attached/plugged in.)
6. Now put the battery back in.
If this doesn't help, then test the power cord to see if it is good. Either try your cord in another device, or try another devices cord in your laptop.
Should that not be the problem, then check the battery to see if it needs replacing.
While I understand you feel the battery is 'built in', it actually should be removable. It may be very complicated to remove (that does happen) but they are removable. Here are two videos on such a device and the battery removal. https/youtu.be/IGAO6V5dfcc and https/youtu.be/egbkSCGRbLc
And here is a page (with images) on how to take it apart. http/laptopmedia.com/highlights/inside-lenovo-yoga-500-disassembly-internal-photos-and-upgrade-options/
Now if you don't feel comfortable trying to do the removal yourself, or if the device is under warranty and you shouldn't open it up yourself, then I would contact the manufacturer about the issue.