I recommend going to Manage Your Content and Devices. At the top of the page, just under the GO button for the search box, there's a link to Kindle Support. It will show you icons of any Kindles you have registered. Click on the one you are having trouble with. On the next page there will be a search box, either at the top or the bottom. Type REPLACE BATTERY into it. You'll get a list of several help topics. Choose the one with a title similar to CTIA 1725 Battery Safety Statements . (The letters and number might be different). I have a Kindle Touch and a Kindle Fire 8.9. Kindle Support's information for both of those says batteries should only be changed by an authorized Amazon service provider. I don't know about any of the other Kindles.
I seriously doubt it is the battery. That particular model (and several others) are subject to solder connection problems with the power jack (inside the unit). This is a very common problem. There are several people on the internet that repair them at a cost of about $45.00 Should you be handy with a soldering iron the repair kit can be purchased for about $10.00. I've done one here at home and trust me, it is well worth the $45. to have it done. A quick search of google will bring several options.
If the connector is a mini-usb, try using a different charger. That type of connector doesn't seem to last very well, and stops making good contact after a while. If you have a phone charger that works reliably, try that.
After many hours charging the screen came good enough to actually read pages from a book but a battery warning comes on to tell you to charge up, and the battery icon top-right simply has a ! mark on it.
Batteries on the first and second generation Kindles are dying for a lot of people by now, several of which break the case trying to get them apart to change the battery. As a formatter, I'd be tickled pink except for the fact that Amazon is still selling the DX new (which is a 2nd gen Kindle device).
~~~ eBook Pioneers (A formatting service. I format in HTML to have more control over the fine details.)
Making eBooks With Jutoh (A free Jutoh training movie for people who want to tackle formatting their own books but don't want to learn HTML/CSS.)
"When you have got quite alone, sit down and be lonely ... fold your hands in your lap, and be still. Do not try to think anything ... by and by, it may be, you will begin to know something of nature. Nature will soon speak to you, or not until, as Henry Vaughn says, some veil be broken in you." ~ George MacDonald