Often these were ripped off at the roof and sealed over, then sealed off in the kitchen ... you never know until you get in the attic and look.

"I was thinking of mounting the hood and running duct to the opening and sealing it with some type of calking to seal any gaps? "

Really BAD idea ... you need an exhaust vent that goes all the way out to the exterior of the house, with all the joints sealed with metal tape.

Your plan would exhaust wet, greasy air into the attic.

I have seen them nicely retrofitted, but they installed the fan at the top of the hood area and a proper vent pipe.

I am on the first floor of a 2 story building. I asked the landlord if she would have the vent inspected with no response. The city inspector could care less when I asked them about the requirements. I need to do something since my kitchen and living room fill with smoke and the smoke alarms go off every week. The best I can do is test with a smoking gun close to the opening to see if it draws up smoke.

assuming the vent is open, what is the best way to tie into the existing vent.

First: Look at the roof over your kitchen area and see if there is a vent cap over that area.

If there is a cap, you have some possibility of having a clear exhaust route.

Buy, borrow, or rent an "inspection camera" and look up there to see what is there. they are the same fiberoptic camera style used in medicine, but for looking inside soffits and attics.

You should see a clear tube or square channel, with some light coming in at the top.

If you do not see a vent going all the way up and out, do NOT vent your kitchen inside the building.

Here is a link that might be useful: inspection camera