It is hard for me to guess what is going on. Even when the password is wrong, it
will prompt you again. It sounds like more is going on than broken dpkg. Try
some other commands that require sudo and can do no harm such as: sudo lshw. It
will show your hardware. This just lets you know if sudo is working. Another one
to try is: sudo su. It should change the prompt from a $ to a # sign, meaning
that you are now root and any further commands will be executed as root. This is
useful if you don't want to type sudo before each root command. None of this
resolves your problem. It just let's you know two things: one is that your
password works and two is that sudo works.

Fixing broken dependencies is complex. It is helpful if you can remember the
last thing that you installed before it broke. The reason is that it could be
from apt or dpkg and if you know the programme you can try to remove it. If you
can't remember try in order the following:

Open Synaptic (in the Menu under System | Administration. In Synaptic go to
Custom Filters in the bottom left. Choose Broken. If there is a broken apt
package it will show here. You can right click on it and choose Mark for
Complete Removal. Then Apply the changes.

If this does not work open a terminal and type: sudo apt-get -f install
Next comes where it is helpful if you knwo the name of the package: sudo apt-get
remove --purge <packagename> with the name of the pagage replacing the brackets
and the words in between such as sudo apt-get remove --purge gedit will remove
gedit.

Afterwards type: apt-get clean

Failing this you can try: sudo dpkg --purge <packagename>

If you don't know the package name then try: sudo dpkg-reconfigure -a

Hope this helps. I have used these to fix my apt/dpkg , but am am firing blanks
because I can't see your system and the output of the commands.