Broke dryer hose… need advice

Sorry to be the Nay Sayer here, but trying to repair a 240VAC heating coil element is just plain dangerous. You mentioned using aluminum wire, WRONG! That wire will melt in a heartbeat as soon as you turn it on. The heating element is probably made of tungsten or other hardened heat producing metals and trying to use copper or aluminum is no substitute.

Don't risk burning up other parts of your dryer or even your entire house. Order the correct part or go pick one up at a local appliance repair shop and do the job properly and safely.

I'm presuming you're using the pillow case as a lint filter. The bigger problem will be humidity. You'll be putting all of the moisture from the wet clothes into the air. After a couple loads, this makes the room really, really, REALLY humid (I've been there).

If you can swing it, you might be well off getting some flex dryer duct and running the end out the window while you're doing laundry. Then you won't need the pillowcase at all, which might cause some other problems due to back pressure/poor airflow (e.g., it might take longer for your clothes to dry, and it might cause the dryer to run hotter).