I think other are saying to possibly use rectangular "inside" the wall behind this cabinet that is between your kitchen and DR. If that is the case, I do not think that will work or is necessary. 1st you would have to bend/transition to get that into the wall cavity. You want to eliminate bends as much as possible if you can. Not to mention you will have to take a nice chunk out of the back of the cabinet to accomplish this I would think. Not to mention I think you need something like a 5x12 rectangular duct (narrowest) for the same cubic foot of air movement for an 8" round (according to charts I have seen on the web). This will not fit in the wall. If I am looking at the 2nd photo in your attic correctly, you are saying on the right side on the photo, above the electrical junction box, that is mounted on the joist, is where the hole is located. There is another joist behind it that is the attachment point for the DR ceiling Drywall. There is block on top of the joists connecting the two and there also appears to be horizontal blocking between those 2 joists as well. So again not sure anything could be run inside the wall. If the walls are 2x6 then you have a shot. You said you have 6" from the edge of the joist to the front of the cabinet. So you may be able to transition something here within this space. I brought up Oval duct since it has much better/smoother flow, compared to Rectangular. Also, it would be 3 1/2" width and you should have about 3" now, from the front edge of the circle to where the wire was hanging down. Thus, you would only need another 1/2" of cabinet cut to fit the duct towards the front. But the duct would require you cutting wider.

Is the attic space accessible? If so, a qualified carpenter should be able to climb up there and assess the situation. I'm not a professional, but if it were me I'd call my local lumber yard (not a big box store) and ask for referrals of Home Remodel contractors. You need a carpenter to fix the problem, not an installer. The link below will give you an idea of what the carpenter may be able to do so the ducting can be run. Here is a link that might be useful: joists

This configuration should be engineered by a competent architect. (If you are a mechanical engineer, you might be able to do it yourself.) In essence, the ends of the new suitably strong "lighter brown" beams that the cut joists mount to have to be supported at each end by structure capable of handing the increased load at each end. E.g., the joist just beyond (to the left of) the cut shown will need to be reinforced. The other ends may be attached to the wall. Usually, Microlam® like beams can perform the necessary horizontal support for the joists. The wall mount may require wall strengthening or columns in the wall.

Ok then, so what I saw on that specific joist has a 3rd "top plate" which was cut around it. If they cut it close, then that means the rafter tie may have moved out leaving the open notch. Or they gave themself a lot of play in the cut. If your rafter ties are just in tension, it is not ultimately critical that you are shy of 1 1/2" bearing, but it would be better and code, and it looks like the ones pictured all fall short. Especially if you are storing things or crouching up there on them, you should meet the 1 1/2" bearing. As far as stabilizing them from moving more, you could sister them with full length joists if you choose, but at that point you might as well remove what's there and size new ones for storage. Alternatively, all you really need to do is resist the thrust of a butt end. So you could place a 2x4 flat across 2 house ceiling joists spaces, then another turned on it's side fastened to this, then extended say 32" sistered next to the rafter tie. It all depends on how messy the house ceiling framing is and what is in the same plane as each other. What might instead determine how you fix the framing is how you are going to fire separate the garage for life safety. You need to do it. You either need to put 1/2" drywall and make a garage ceiling, or you need 1/2" drywall on a partition wall in line with the house wall extended above the attic all the way to the undersisde of roof (local AHJ may have stricter Type 'X' addendum). I am guessing that the size of rafter ties (2x6?) are to try and span the garage width, I do not know how they frame into the rafters on the short hip side though. I'd bet that there are no kickers on either long side hip down to these rafter ties either, if so, since these are perpendicular to the house ceiling joists, the long side hip ridge has probably sunk, the side wall has pushed out slightly (3/4" may be imperceptable to your eye), the 2x6's have sagged slightly, and all that together has pulled these rafter ties away from the house wall since they only had 2 toenails in them.