let's assume for a second that gore won his recount in Florida, 8 months in to his presidency, the twin towers came down. how would have gore responded?
Probably the same way, initially, with an invasion of Afghanistan. It's not clear whether that would lead to a capture or killiing of OBL at Tora Bora. Probably not. The reason for the failure was the use of natives to guide Special Forces, who were untrustworthy. Those same guides would still be there.
Thereafter, there'd be continued interest in taking the war to Pakistan, rather than shifting interest to Iraq. This would put pressure on Pervez Musharaff, possibly causing his premature collapse and replacement by a democratic government. It is unlikely that it would allow US forces to operate in Pakistan freely, and so OBL would still remain at large.
The rest of his presidency would be focused on domestic security. I suspect there would still be something like the Dept of Homeland Security, but if there was a Patriot Act, it is unlikely it would be as drastic or last as long.
Gore was always into environmental concerns, so he would likely pursue those in office, pressing for ratification of the Kyoto Protocol. What is certain is that he would not win his Nobel Prize, at least not when he has.
Without Iraq to cast a shadow over his presidency, support for the Democratic party would persist into the election of 2008, likely resulting in victory for Hillary Clinton as president.