My ds is nearly five months.
Yes, it's defibnatly impossible to entertain them all day. I think dragging moving them round talking to them while you do what you need to is perfectly adiquate entertainment for most of the time tbh!
mine is happy with this for the most part, but yes, he too demands likes some interaction. I do, like you, sing and chat. I have all sorts of songs made up for the poor thing, and classics with actions that I can move him to do, ie heads shoulders knees and toes etc.
He can now grab and handle things, toys ect, and that entertains him quite a bit. Even a muslin or bit of paper (when supervised) can do the trick...it's all new to them isn't it!
He's just got in to rolling, which is great for him, though requires more of my attention! i'm sure your dd will add this to her repertoire soon if she hasn't already!
The other great thing we have is a door bouncer, he's just got the hang of it and is loving it. We picked it up from a charity shop for a tenner and it's definatly a worth while buy (on the contrary we were given a bumbo seat, which is ok, but I can't see it being of use for long and wouldn''t pay full price for it.)
Oh, also we've been given a walker thing which is quite good as he's upright but can't quite yet move himself around in it!
ANd he likes being thrown up in the air etc (vertigo pay is it's posh name )
So I say don't worry too much. You have to get on with household tasks, he doesn't yet know this isn't playing! We definatly shuffle ds round from one thing to another as he gets board. We have times where we play with him intensly one on one, but also lots of times where he has to sort of entertain himself a bit, that's life!