Try the simplest fix first
If the broken key is still near the front of the lock, your first move should be to make the cylinder less stubborn. A tiny amount of dry lock lubricant can help. PTFE-based lock lubricants are great for locks, and only a very light application is needed. Its dry formula is specifically meant to lubricate without attracting the dirt and dust that can make a lock feel gritty in the first place. In other words, this is not the moment to drown the keyway in random oily spray and hope for the best. Use a little, give it a moment, and keep your movements gentle.