Ignoring the part about the discount factor depending on human capital, and switching to finite instead of infinite horizon, the model essentially becomes the same core setup as used in Life Cycle Model 42 (in the Intro to Life Cycle Models).
Beyond this the toolkit cannot presently solve the Ludwig & Schon (2018) model. [It cannot do the experienceasset in infinite horizon, and cannot do that the discount factor is a function of state h.]
Link to health risk post Alessandro mentioned: Age-dependent health shocks
As he said, it is a loosely similar setup, and fits with what toolkit can do.