"Among other things, the minority has no incentive to agree, and the majority can't let debate go on forever, because they have to pass bills to keep the country going."So give them an incentive; get people to the table and come up with a compromise that has the support of enough people to vote cloture on any filibuster attempt. The filibuster, cloture and other rules should force compromise and create policies in the best interests of the nation as a whole. As Kennedy wrote, "Going along means more than just good fellowship - it includes the use of compromise, the sense of all things possible... For politics and legislation are not matters for inflexible principles or unattainable ideals" (Profiles in Courage, 4).
Getting rid of the filibuster would more than likely have a negative effect as the rights of the minority would be consecutively overruled. I don't think that's a path either party wants to go down.
For some background on the filibuster, you can go here.
The longest filibuster in Congress was by Strom Thurmond on the Civil Rights Act of 1957; it lasted for 24 hours and 18 minutes.
According the tour guides at the Texas Capitol, the longest filibuster was 44 hours long in 1977, but the Handbook of Texas Online states the longest filibuster in Texas history occurred in 1957 by Henry B. Gonzales and Abraham Kazen, lasting 36 hours.