What should we do after entering Mecca with the intention of Hajj and performing Ihram from Al-Taneem, then performing Umrah and departing without completing Hajj due to the illness of a family member?

You must repent to Allah Almighty for your negligence in matters of Sharia and your shortcomings in learning them.

Leaving the state of ihram from the miqat necessitates a compensatory sacrifice (dam) to be slaughtered in Makkah and distributed among its poor. If one is unable to offer the sacrifice, then he must fast for ten days.

If you entered ihram for Umrah as mutamatti' (performing Umrah then Hajj), and left Makkah before entering ihram for Hajj, then there is nothing upon you.

However, if you entered ihram for Hajj (either ifrad or qiran) and left Makkah, you remain in the state of ihram. You must return to Makkah to exit ihram by performing Umrah, and you must perform Hajj the following year along with a sacrificial animal (hady) due to your negligence in missing Hajj.

If you are unable to return to Makkah, then you must exit ihram by slaughtering a sacrificial animal, similar to one who is prevented from completing the pilgrimage (muhsar).

If you entered ihram for Qiran (combining Hajj and Umrah) then converted your Hajj into an Umrah, exited ihram, and departed, you have acted improperly. However, you are not required to make up the pilgrimage or offer a sacrifice according to some scholars who permit converting Hajj into Umrah.