Dr Hamad bin Ibrahim Al Haydari says:
It is not permissible to eat food that has been cooked with meat that has not been slaughtered, according to the rulings of the Islamic law, because the meat at that point is unclean, Allah SWA says: (Say: ‘I find nothing in what has been revealed to me that forbids any one to eat of any food except the dead, running blood, and the flesh of swine – for these are unclean – and that which has been hallowed in its slaughter to other than Allah) Al An’am: 145.
At the time when the prohibition of donkey meat was revealed on the Prophet ﷺ said, he ordered the pots which the meat was placed to turn them upside down [see, Al Bukhari: 2991, and Muslim 1937]. So, if the broth was permissible to drink he would have thrown the meat and kept the broth, and know that Islam does not forbid a food or anything except for its dirtiness and its harm to humans, as Allah SWA says about His Prophet ﷺ: (and who makes lawful for them all the pure and good things, and makes unlawful all the impure and bad things) Al A’raf: 157. But if it is meat that has been slaughtered in a non proper way, because the way it was slaughtered was against the Islamic law then its broth is also forbidden.