Learn Turkish: Mademki bu problem seni ilgilendirmiyor, lütfen sus. - Seeing that this problem does not concern you, please be quiet.

You can absolutely use madem here with no change in meaning. The suffix -ki is a Persian loanword addition that permanently attached itself to several conjunctions over time (like belki, çünkü, and sanki). For madem versus mademki, both mean 'since' or 'seeing that' and are completely interchangeable.

Why does ilgilendirmiyor take the accusative seni instead of seninle?

The base verb ilgilenmek (to be interested in / to take care of) takes the instrumental case, as in seninle ilgileniyorum (I am taking care of you). However, adding the causative suffix -dir creates ilgilendirmek (to concern / to make it someone's business). This causative version requires the accusative case to show who is being affected, which is why it takes seni here.

How is the long verb ilgilendirmiyor constructed?

It is built from several layers. The root ilgi (noun: interest) becomes ilgilenmek (verb: to be interested) via the -len suffix. Next, the causative -dir makes it ilgilendirmek (to concern). Finally, the negative -mi and present continuous -yor are added to mean 'it does not concern'.