How to demo a partially structural chimney

There is no mention where the cut out is but this should take care of it in any respect, as long as the mid span support is a series of posts and not a purlin... Can you get a 2X6 in there that will give close to a full span? if the material is clean or can be scraped clean, a healthy bead of construction adhesive zig-zagged on the face of the old 2X and screw the 2X6 starting 4 inches from the ends with a pair of screws and 12+/- inches apart in the remainder in pairs also. Use 3 inch drywall screws or 2 3/4 inch SDS screws through the 2X6 into the 2X4. Make sure the screws draw tight the material, pilot if needed with a 3/16" bit.

If you have a mid span support that goes to good bearing through to the foundation, you can use a shorter repair joist, but a single joist that goes over multiple spans is stronger than a single span joist.

FWIW, it looks like a 2X4 will suffice from what I seen the table at page 396 of the 2012 building code (table R802.5.1).

Since you are redoing the roof deck, you are at an ideal time to fix this right, or as right as you can without redoing the major framing.

Depending whether you want the water to all go one direction, or go two ways, you build a flat (but sloping) triangular roof in the valley - either raised at least one foot and sloping all the way to one end, or raised at least 6" in the middle and sloping both ways.Cover with ice and water shield (first) and metal flashing in a wide continuous sheet running up both roof slopes. Essentially, you are giving this valley floor some slope - to do so you have to give it some width as it rises from where it is now.