It just had a bad current that went through it the player shouldn't fry, check the battery voltage see if its still got the original voltage if not it will be under its capacity.
Rejuvenate the battery and use the right power supply this time.
If its still under warranty take the player and its original charger back to where you got the player from and they might swap it out.
well, it would be really easy to take off the battery. The battery is simply plugued with a little 3-pin wire, u just pull on it and put it back in. Now to open that gigabeat, its pretty easy. I even got some images for u. Go at this website (its for the F10 and u have the F40, but its the same thing):
The manual is here
The connection for external speakers like you have is called analog audio output and it is shown with a little set of headphones next to it. The issue is the size of that jack. There are two basic sizes 3.5 mm and 2.5 mm . The manual does not say which one is on your television. If you have a set of headphones you can try that would let you know. Otherwise it is trial and error. What you need to do is get a cable that has that end on it and then make the connection to your speakers. Using the wires on the speakers you need to be able to tell what polarity they are and connect two of them to one of the wires in the cable that you found that fits the jack. These are the ground connection and on the cable the ground is usually a bare wire strand. Then the other two cables that are remaining, connect to each of the wires that are coming out of the connection to the television ( they should be colored) One goes to each for left and the other for right. You may want to solder them and make sure that you tape or insulate them against shorts.