I found out that an ingredient in my cleansing balm is donkey milk. Is it halal to use for Muslims?
As there's no reason why one should think donkey milk is haram at all. It is perfectly halal, AFAIK there's no madhab/sect which even quotes donkey as an impure animal.
Those animals which are often discussed in this context are dogs and pork. Also note that what exactly is discussed in this context is: their flesh (we clearly know by Quran that it is haram to eat pork, while eating dog is haram by hadith, but here the authenticity might be discussed), their skin/pelt and spittle. On these issues there are different opinions except for what is clearly haram.
As you neither eat nor drink this milk and there's no evidence of donkey being considered as haram, impure or anything which might lead to the conclusion that using their milk in a skin care product might be considered as unlawful, there's no reason to think of it being haram. This is based on the fiqh rule saying:
الأصل في الاشياء الاباحة, ولا تحريم إلا بنص
Meaning: By default anything is allowed or permitted (halal, mobah) nothing is forbidden unless we have a -clear- text (Quran, sunnah) saying so!