Our kitchen sink is draining into our laundry sink and the laundry sick will not drain or at least extremely slowly! Please help!

DIid you plunge it at all? I would remove the laundry sink drain trap under the sink and check that first. If that's good then maybe a clog down the line. The only thing left is the plumbing there is not vented

Just a guess, kitchen sink is located directly above laundry sink? You will need to auger drain from laundry sink down. There may be a cleanout for this purpose. Look around and see if you can find a cleanout for this drain. If not, you can unhook trap from laundry sink and snake it from there. Make sure to put in enough snake to reach house drain. Good luck and let me know how things work out.

Ps. You may also be able to snake from kitchen sink vent on the roof.

The problem you are experiencing is common. Please keep us posted and we will give you advice as to how to solve it.