Well, there is a few things that could be wrong. First of all, is the battery space (I can't think of the actual name for some reason ) corroded? If so, you'll need to gently clean it and that will (hopefully) make it work again. However, if the battery space is not corroded then perhaps you left it in the sunlight for to long? I do not quite know how to reverse the effects of that one... And my final guess (if all the others are incorrect) is that maybe you got some water inside of it? You can (possibly) reverse the damage of that by putting your tama in a bag of rice for awhile.
That's all I've got! I wish you luck on fixing your iD L. ^_^