Allow users to ssh into pods through a single service in k8s

Question

I am developing an application where users can spin up compute pods running jupyter notebooks and can ssh into their pod. We are using k8s to provision and manage the compute. Since we run our infrastructure in AWS using EKS and elastic IPs are scarce, we need to route ssh traffic through a bastion instance which forwards ssh traffic (also http for jupyter notebooks) to the correct pod. I am hoping for any suggestions on how to implement this. From my understanding so far, I need to have a separate port for each user for ssh on the bastion instance. This seems unwieldily but AFAIK ssh traffic cannot be routed in any other way. For HTTP, we can have routing rules which should be much more straightforward.