Summary:
At highest point:
Kinetic energy = 0
Gravitational potential energy = mgh
Elastic potential energy = 0
Total energy = mgh = λ(e + x)²/(2a)

When string just becomes slack:
Kinetic energy = ½ mv²
Gravitational potential energy = mg(e + x)
Elastic potential energy = 0
Total energy = ½ mv² + mg(e+x) = λ(e + x)²/(2a)

At equilibrium position:
Kinetic energy = ½ mv²
Gravitational potential energy = mgx
Elastic potential energy = λe²/(2a)
Total energy = ½ mv² + mgx+ λe² / (2a) = λ(e + x)²/(2a)

At lowest point:
Kinetic energy = 0
Gravitational potential energy = 0
Elastic potential energy = λ(e + x)²/(2a)
Total energy = λ(e + x)²/(2a)