Forgot passcode and home button is broken, how to I initiate restore

You can try powering it off. Then launch iTunes in your computer and have it open and active. Now plug the phone into your computer’s USB port and see if it boots into recovery mode and allows an iTunes restore.

However, without a working home button you’re just going to encounter further problems using it. The home button is an integral and necessary functional control element of every iOS device. Best to take it to Apple and just get it fixed.