Short answer: Use Alt8969 (important: type in the number on the numpad, while holding down the "Alt"-key).
Long answer (and limitations to the solution): In my test this works with the Win7 password logon input field at the logon prompt (but of course you won't see the character in the password field). I created an account with a password including that character (by copy&paste) and then logged on using the new account successfully (entering the character with my solution). It's possible to replace the native logon prompt, for example with third party software, in this case my solution might not work (if the new logon prompt can not handle Unicode characters). What I haven't tested yet is if the solution works for different a different system code pages.
If you want to "see" that it works you can do it, for example, in Win7 MS Wordpad (use e.g. Cambria Math as font). Wordpad will accept this input method as well as display the character character correctly.
It does not work, for example:
if you use this input method with MS notepad. Notepad will process the character by using copy and past (although it shows a different character, the binary code representing the character is correct) but not process it correctly with the input method (it creates a different character on a binary level)
if you use it in text file that uses the ANSI format (instead of Unicode)
This type of character is usually not used for a password that has been created automatic. Usually only characters are used that can be typed by a regular keyboard are used. So I would also make sure that it's not a different character for example a funny looking "7" (seven).
If you ultimate goal is to change the password and you don't insist changing the password by typing in the current password first (which leads you to your problem), I would consider to use a workaround. For example if you have a different account (with the right to change passwords for other users) just log on with this account and change the password.
Just to "finish" the question: according to Ehouarn the final solution was to ask an admin to change the accounts password (who of course has an account with the right to changes the password your account).
Furthermore, this solutions did not work for Windows 10 as this Test by Ben N shows: in the Windows 10 password prompt (which has a button to show the password as typed so far), you get ○, U+25CB WHITE CIRCLE.