Consumer surplus in case of perfectly inelastic demand
From a purely theoretical perspective, if an individual's demand curve is perfectly inelastic, then her willingness to pay for the good is infinite. NB this also implies that she has an infinite budget. Thus, consumer surplus is well defined: it is the willingness to pay minus the price she pays, so as long as the price is finite her consumer surplus is finite.

In practice, no one has an infinite budget. So if the individual's demand curve is truly perfectly inelastic (i.e. the inverse demand is vertical), there exists a price such that beyond that price she can no longer afford to buy the good. This price is her willingness to pay, so consumer surplus is again well defined: the willingness to pay minus the price.