What is the ruling on dealing with lawful money mixed with unlawful money, and how and to what extent should one dispose of the unlawful portion when it is impossible to differentiate between them?

You have done well by changing your field of work, for internet cafes are often not free of what you mentioned. The money you earned previously from working in the internet cafe is permissible for you, because the contract was based on benefiting from the service, and the default ruling for that is permissibility unless otherwise known.

As for writing research papers for students and submitting them ready-made to them, this is not permissible due to the deception and forgery involved. However, if you were ignorant of its prohibition when you earned the money, then it is not forbidden for you to benefit from it, according to the Almighty's saying: "So whoever receives an admonition from his Lord and desists, he may have what has already passed, and his affair is up to Allah."

However, if your money is mixed with illicit gains and you do not know its amount, then you must spend the amount you are certain is illicit in charitable causes.