I might not be any help - but here goes.
When you say she has been acting broody - exactly what do you mean?
A broody hen: will fluff up and growl at someone approaching the nest. She stays on night and day when off for a short bit, she eats/drinks potties and goes right back to the nest. When she walk her walk will be normal but she may or may not fluff up and stick her wings out (appearing bigger) and be grumpy when approached by people/chickens. A broody
does not lay eggs.
A sick or ill bird often will also stay in the nest box because it can be safe from other birds there. Will not protest when you are near the nest. When walking will often look wrong tail will droop, chicken might walk and then lay down again. An ill bird might lay.
The egg I am guessing was either an egg bound egg that someone expelled quite possibly your broody girl, or a meat spot egg. With the smell you describe, I am guessing/ leaning more toward egg bound. This site has lots of information on eggs, look to the left side of the page for more information links:
http://www.thepoultrysite.com/publications/1/egg-quality-handbook/29/meat-spots
There are many, many threads on BYC dealing with egg bound hens the easiest way is for you to search and read about the possible helps. To search use the BYC search link which is found in the upper blue bar, just above your logged in name NOT the google search. BYC search will allow you to search the words egg bound in the emergencies thread. You can also search by author Speckledhen, unfortunately, has dealt with this issue and has lots of good advice.
I wish I could be more help. Broody experience I have, egg bound experience not yet, but it is just a matter of time.
for you and your girl.