Galaxy tab2 has no HDMI connection. If you want to watch recorded video and pictures on TV you will a) need to down All Share app from Google Play (also called Samsung Link) to the Tab 2 and b) have a TV that supports DNLA connectivity. You then use All Share to play video on that TV through Wi Fi over DNLA.
First check if all contacts, on the battery and in the tablet are clean and shiny. Only clean contacts with a dry cloth or cordon swab.
Just a scrap of paper or some dust can spoil all the fun.
If this also does not work you can do what you said, goto the repair shop, or buy a new battery.
Replacing the battery is trivial once you manage to remove the Galaxy Tab's back cover. However, that is the catch: the back cover snaps on the case using roughly twenty rather fragile plastic fins and it is very easy to break them off in the process.
I do not recommend to attempt this without some experience and proper tools - forget about prying the case open with a screwdriver, you'll nick the plastic badly and you may break off the lot of the fins.
If you're going for it: you have been warned. You will need to order a battery replacement kit, you can find them, for example at Amazon, for about USD 50 plus shipping.
Watch this video: Samsung Galaxy Tab 2 7 0 Battery Replacement for the opening procedure and the kind of tool you need, note how careful the presenter acts and how it takes even this expert several minutes to remove the cover without breaking it. I also assume the case used in the video was opened and closed before and by slight wear, it comes off more easily compared to a virgin immaculate case.
The actual battery replacement is done by removing six screws and a connector and after changing the battery, putting the parts together in reverse order.
Try holding down the button on corner end and at same time hold the volume up button, keep both depressed for about a minute and should reboot.
If that does not work try the corner button and volume DOWN button for the same minute timeframe.
Take out its battery and re-insert it after 10 minutes. This must make it a hard reset. If it do not help you, contact the service center. Gopakumar Gopalan Google