Beth, I've got a better understanding now of the situation; I'd first thought your enclosure would attach directly to the house.
I can think of 2 methods to handle it; an enclosed ramp (like a miniature of what you would board a plane on), or an enclosed "stairwell."
Form your window, you could build a "box tower" straight down the side of the house, with a series of ramps inside, then make the remaining trek to the enclosure on the ground in a wire causeway. This would allow you to utilize the house for one side of the enclosure, and the remainder could be covered in siding to match or compliment the house. I've seen some carpeted multi-level "kitty condo's" which had ramps with short jumps between levels, and this is kind of the idea I'm speaking of.
As to an enclosed ramp, that's also possible, and maybe the easiest, most durable way to do this would be to have a ramp fabricated for you from galvanized sheet metal. Not necessarily a complete enclosure, but similar to a playground slide, which you could glue a runner of outdoor carpeting on for traction and then wire enclose the sides and top. Depending on the length you needed, you might even be able to simply buy a slide to use for this. Your wire enclosure could be attached to ribs (similar to a covered wagon) made from 1/2" electrical conduit bent to a "U" shape, 3/4" pvc pipe w/ 90 degree elbows; any number of things like that.
I'm excited by the possibilities already, Beth, let me know if you open it up for guests; I've got 3 who would LOVE to make reservations!
If you need further detail or assistance, we'll be here! Good luck with your interesting project-