Hm, thas very much possible.
I thought I could start the io_service object again, if there is new need for it.

You can restart io_service::run after it returns, but you should call io_service::reset() first.

Maybe I should make a Session class, which holds its own io object and inputhandler, so I only create a session in my client, and the rest is then tied to the Session instance. Would that work?

The decision who should own io_service(s) depends on your application design. You've got examples of asio-based design here:
http://www.boost.org/doc/libs/1_38_0/doc/html/boost_asio/examples.html