Elgg is not a library, it's a framework. Both Elgg and Laravel have their opinionated APIs and won't work together, not unless you rewrite Elgg's routing system, database model and queries, view system etc.

Thank you! 

juho and others are partway through coding a full plugin solution to do this and also push chat messages. i don't know how far they have got.

yep: https://github.com/juho-jaakkola/elgg-live_chat

If you are not using Shared Hosting, then  You can use Websocket interface by running an Separate chat process(out of apache). I have implemented it. Its  just 100 lines of code. Any other solutions are resource hungery because it goes via Web-Server(Apache/Nginx). and Web Server is not meant for live connection.

I mean how to pass data from client side to the server. Should I use JSON, XML, or something else? It seems the exposed function is not able to process JSON data.

I found the solution. The web service returns requested data in the form of JSON but it can not process JSON data sent from client side. Instead we should pass URL encoded data to the server.