Well it sounds like the device hasn't charged/isn't charging and that is why it isn't turning on.
First you need to make sure your cord is actually working. If you can try using it on another device, or try using a different cord on your tablet. If the cord works on another device, then your device is the problem. If another cord works on your device, you know the old cord is the issue.
Also you can try checking the button to see if it is stuck. The outer button is just a kind of cover for the actually inner button that you are pressing down. If the button is stuck, it won't let you turn it on and it probably wouldn't charge either. You can try checking this while the tablet is closed (pressing the outer button a lot to see if loosens up) or you can open the tablet and see if you can press the interior button and get it to load.
If it won't charge, even with a new cord, and the button doesn't seem stuck, then the device is the issue. It could be the battery, which you would then need to replace, or it could be the port where the charger plugs in. If it is either of these, and you feel comfortable doing it yourself great. If not, I suggest taking it to a tech to have them do it. Especially with replacing a port (if that is the issue) as it is usually attached to the motherboard, and then that would need to be replaced, if the port is the issue.
Start with the cords.