I logged a case with MS Support and when they came back to me, they advised that the service principal that is spun up in the background had expired. This service principal is required to allow the cluster to interact with the Azure APIs in order to create other Azure resources.
When a service is created within AKS with a type of LoadBalancer, a Load Balancer is created in the background which provides the external IP I was waiting on to allow me to connect to the cluster.
Because this principal had expired, the cluster was unable to create the Load Balancer and the external IP of the service remained in the pending state.
There were a lot of steps here; click through to see just how many.