I never use auto, I've been doing all of this in manual mode.
This just isn't true with my D5100. When my flash is open in manual mode it doesn't always fire. It will be firing fine and then when I changed my perspective slightly it will stop firing completely (even when it's open). I don't know if my camera is bugged or what. I've searched all over the internet and posted on several forums and no one seems to have an answer. I've never had this problem indoors, only outdoors. It's refusing to fire in certain conditions and I've found no way to override this no matter what I try.
OK, sorry then, I will back out. I do not know about that situation. I don't have a D5100, but all my Nikon cameras work as I stated (internal flash fires if door is open).
If you are firing again too quickly, the internal flash will NOT fire if it has not recycled Ready. The Ready LED has to be ON to fire. But if you are waiting 2 or 3 seconds for the Ready LED to be On, my opinion is that it will fire if the internal door is open.
Sorry if this is no help.