Now I don't entirely know, but sometimes when I try and log in to servers I get the invalid session message. When that happens I need to close minecraft and the launcher down before re-opening it.
I would also possibly suggest trying to connect by the Direct Connect option. Get your brother to search on his computer for the application called Command. In command type in ipconfig and it should come out with a load of text on one side and numbers on the other. Look on the left for the IPv4 address and record the number that it says on the right. Then, in direct connect, type that number, with the dots, the a colon, and then the 5 digits it put in your brother's chat when he opened it to LAN. Keep in mind that the IPv4 address can change sometimes.
Other than these, I don't know what to suggest, but I'd give them both a try. If they don't work I hope someone more knowledgeable can help you out!
Have you tried using your email as the username? It's silly to say this but also make sure caps lock isn't on. If that doesn't work, I would just reset your password.
CookieMan2 got a reaction from Lord Ptolemy in Can't Connect Via LAN