I know I am very late to the thread, but I have a D5300 as well. As has already been noted, the battery has likely swollen and should be replaced - but you still need to get the current one out.
The D5300 has a rubber tab that can be pulled aside to insert a dummy battery for external power. This provides good access for a narrow (e.g. #3 std blade) screwdriver to help the battery out from the side. Off course the battery lock tab needs to be pulled back at the same time. Hope this helps.