I live in Maine so totally understand the freezing weather. I can clean inside out though so don't have the same issue, however, I would recommend a long flexible hose, connect a few together if you must - hook up to hot water, hose down as you would normally and then squeegie excess water because as it freezes, it will become like an ice rink.
I have the big thick freezer curtains that I put up around my outdoor runs during the winter to help keep the windchill out and snow blowing in and to keep the temperature a bit warmer than outside, and when the water is freezing as soon as it hits the ground, I have a torpedo blower that I have on to dry the floors of somewhat to make it less slippy.
Something else I do - is I have a floor scraper, a large one and I will just scrape up the frozen pee / poop and put in bags to throw in dumpster - that is when everything gets frozen solid.
Winter isn't fun for cleaning outside runs for sure - good luck.
Jen
------------------------------
Jenny Cope - Fear Free Certified
Exec. Dir./Founder
Give a Dog a Home Rescue
Greater Goods Charities - GOODS Program Ambassador
501c3 Non- Profit Corp, Tax ID/EIN 27-5241306
State of Maine Shelter/Rescue Licence # F1463
State of Maine Registered Charity: CO11334
https://giveadogahome.rescuegroups.org/------------------------------