The white plastic tit that sticks out the side of the stalk is the bit that makes contact with the cam on the steering wheel that turns it all off after the wheel returns back to straight ahead. Try taking the stalk off the column and pushing on it after you pull the arm to the turn position to see if the arm returns.
The internals of the indicator stalk could be either worn or has a part broken or a spring popped off.You can take the stalk assembly apart and pull them apart if youre careful check the internals for this but take note where everything goes because its a bit of a jigsaw puzzle to get back together.
From my experience with them it will probably mean a new stalk however.
