What size laminated beam should I use for a 15 ft. span supporting the back wall of a house as well as an extension with a shed roof thank you

you need an engineer for that, if you don't know one contact your inspection office and they can lead you in the right direction,
keeping in mind im not and engineer , what i would say is at least an 16" X 2" beam and you'll need 2 of them. I don't think you'll be made to go any bigger than that.
you'll need to hand drive some 16ds to fasten them i believe.
also you need to cut out the sheetrock in the wall and add 4 2x4s on each end of the beam for support.
i don't know if you've ever put these up but don't try this without help. And i don't mean you and a buddy lol. At least 3guys if not 4.
good luck to you