It should not take that long for it to complete the boot. That might be one indication of a problem.
Have you done anything to modify the log in process? I am not used to seeing the lock screen you show, but it may be normal for you.
If the onscreen keyboard will not start, perhaps a virus or some type of corruption is involved. When you selected the onscreen keyboard, did the computer say anything? It appears you are typing on a layer which sits over the password box. Almost like something is intentionally blocking it.
You might try removing the password requirement from the boot, just for testing. There really is no reason I can think of where it would cause a problem, but something to check anyway.
If you could get into Safe Mode, you would probably be able to enter your password. It sounds like you don't have any recovery media, are there any Factory specific keys available for starting the system in another mode or allowing you to enter the recovery environment?
Edit: I should also add, if that system has a webcam, be careful until you know more.
Edit 2
: Are you using any type of password manager?