Hello, I can do your project very fast. I am very experienced in developing webapps.
The app will have a Server side, which will manage the users and also be use, if you want, to monitor the chat.
The Client side, will be where the users login and chat.
Would you want to support several chatrooms?
I can also make an android app if you want(also included in the budget).
Please tell me if you are interested.