You just accepted a new job. Congrats!
There's so much excitement surrounding a big career move. Unfortunately, you have to give 60 days notice to your current employer per the terms of your agreement.
The long notice period required in many agreements serves no great purpose, except to make everyone generally miserable. A 30 day notice period should really be the longest period required.
Think about it. If you're the employee, you've already made the decision that you want to be somewhere else. All the projects on which you are working will not receive the same great attention as before. The interactions with the people in the office are awkward at best.
From the employer's perspective, if you think having this person around for an extended period of time helps you in some way, you're fooling yourself. It's hurting you far more than it is helping you. The other employees all begin to wonder what's out there. "Take us with you!"
All those projects can be done by someone else. With a two week, or 30 day, notice period any employer should be able to redistribute the work and figure things out. If not, then perhaps this is one of the reasons the employee is leaving.