Hi! I am a licensed real estate agent in the state of Kansas. Since beginning in real estate September of this year, I have had lots of people ask me if I handle rentals. I know that some agents do, but I have no experience in this other than commercial. My question is multi-facted. First, I want to know if it is legal for me to advertise another agent's rentals. Beyond that, I really don't even know how to approach the other agent when I find someone interested. And what would be a reasonable fee structure for bringing a tenant for a 1 year lease? Any and all information is greatly appreciated!

It's only legal if you have a written agreement with the other agent or an owner if no agent.
How is it that you are able to be an agent without working for a broker? All the states around the Northeast where I live require it.

Often split commissions are 60/40, 60 for the listing broker. But anyone can set their own terms.

I do work for a broker. From what I have seen, a typical rental referral would net roughly $100 per signed lease agreement. So my broker also makes a commission on anything I rent? Meaning I'd see $60 of every $100? (My split is 40/60)

Yes, your broker makes a commission on anything you do that garners income.
You can't do any of this without permission from your broker.
What you get depends totally on what you work out.
Some brokers raise or lower the split depending on how well you do.