New Roof - Part Shingled, Part Rolled Roof

[jc4]

I have to replace to rolled roof over my kitchen. Here's my forum post for the whole story: https://forum.mrmoneymustache.com/real-estate-and-landlording/roof-blown-off-ceiling-collapsed-kitchen-flooded-figuring-it-out/

The kitchen roof is currently 4x 4 foot wide pieces of what looks like a 3/8" -1/4" rubber sheet. It doesn't have the shingle texture to it.

I'm thinking I'll buy a couple rolls of the roofing material at home depot, enough underlayment, new drip edges, and plenty of roof cement to glue it down.

I'll either have to replace 2-3 feet of shingles along two of the sides or may do the whole remaining roof as well.

Are there any better options for the roof? It's very flat. I'm not even sure there's a slope at all.
Would it be reasonable to replace just the kitchen and go back to do the rest later? I have a lot of other work to do, so it'd be nice to postpone the rest of the roof. Thanks Mustacians!

[jc4]

Another question. Someone recomended reattaching the existing roof, then adding the new roof over the top of it. That would add extra weight to keep it on, and insulate a bit more.

Is this worth doing or is that a really dumb idea?