It's not the router that causes lag. Any wireless standard (a/b/g/n) has a maximum speed much faster than the DSi's 2Mbps. Your internet connection will probably be 2Mbps or faster, so that's not the problem either.
We can conclude that it is one of two issues: network latency or server overload. If you are transferring a lot of data in your home network, the router will queue up requests, and the chat data will come through later.
It much more likely that the server takes longer. Imagine 100 or so people accessing your home PC every 10 seconds. That's a good idea of what happens to the server, so that is the reason.
Avatar! Yes, I created the very first DSi Browser extension! (http://dsiextensions.co.cc)