Easiest: replace board.
Medium: replace identifiable components until it works. Mind, you'll need to replace everything that's toasted; likely the controller is dead, maybe the supply to it as well (sometimes there's an aux supply, as simple as a transistor or two, or as fancy as a whole other controller/regulator). Plus whatever caused this to die, which could be anything from overloaded output to bad electrolytics or tin whiskers. (Most SMPS should tolerate a shorted output, but it's a design issue that's easy enough to overlook.)
Hard: trace the circuit, compare with known circuits. Offhand, it looks like it might be a bog standard UC3842 based offline SMPS. I don't get what the two series diodes are doing there, and the driver transistor (likely the resistors and transistor by the gate pin) probably aren't necessary, but eh, it depends, right? Failure could've been various things, including just overheating. '3842 is a current mode controller, safe against overload -- on an immediate (cycle by cycle) basis, but the transistor can still overheat and blow up. It's a control method that solves several ills, but far from every possible one.
Tim