What tools do you have to work with (aside from this magic elf box that will answer any question you ask)?
What materials do you have available?
Which direction are the prevailing winter winds coming from in your area, and which way does the run face now?
What other resources do you have? Church? County Extension office? Family? Friends? A local BYC Club?
Your idea to build a coop onto the back is sound- with just 3 birds*, it does not need to be big: 4'x4' and 4' to 5' high would be plenty, even accounting for chicken math.
A simple coop like that would be easy to predator-proof, and much cheaper than trying to enclose the run you have in hardware cloth.
Some suggestions:
As mentioned above, keep the measurements in 2' increments where possible, to cut down on waste. Build it 2' off the ground and enclose the bottom later on for more run space (shaded, no less).
Use a simple flat, slanted roof and have it overhang front and back. Make the front wall a foot taller than the back and that will give you the slope to shed snow and rain. A scrap piece or two of steel roofing, translucent corrugated fiberglass or corrigated tin would work. The extra foot at the top can be covered in hardware cloth and that'd be all the ventilation that you'd need. One wall is already the back of your current enclosure ...... the floor and other two walls can be made with just two 4'x8' sheets of plywood, and the angle part of the other side wall covered in either scrap or more hadware cloth. Too much ventilation? Tack up some trash bags....
You can often find stuff for free in dumpsters at construction sites (careful, those dumpsters are NASTY).
Check the landfill/town dump, too.
Check the Salvation Army to see if they have a Habitat ReStore anywhere near you- get a window (does not need to open) for a few dollars, and put it on the south side. More light =more eggs, and the sunlight will warm and dry the coop.
If you have a 3/8" drill (corded 3/8" drill can be gotten @ a pawn shop for next to nothing, due to the popularity of cordless drills), then use GRK or Spax screws- they cost a bit more but are self tapping, self countersinking....... faster than nails, and if you mess up or need to change something, they come right out by reversing the drill.
Seal any hole in the roof with a $2 tube of silicone caulking (you don't care what color- whatever is on sale.
Same for paint- Whatever EXTERIOR paint is in the mis-tint bin (you may have to ask about mis-tints), that is what color your coop will be. The coop, particularly the plywood, will last much longer if you paint it.
Anybody else?
* If money is tight, whack that roo- he'll eat more than he's worth, and with just 2 hens, he'll likely ride 'em near to death- it'd be that much harder for them to make it though winter's cold with no feathers on their backs.