Is a Muslim who eats meat not slaughtered according to Islamic law, due to being compelled to do so in a hospital after an accident, sinful, and what is the expiation for that?

As long as you know that the meals provided to you consist of meat from animals slaughtered in a way that violates the Sharia (definition)-compliant method (dhakah), then you have consumed something forbidden and committed a sin for which repentance and seeking forgiveness are incumbent upon you. You could have requested other food or eaten fish and vegetables, unless you were compelled to eat it due to the absence of any other alternative. In that case, there is no blame upon you, as Allah Almighty says: "And He has explained to you in detail what He has forbidden you, excepting that to which you are compelled" [Al-An'am: 119].