Unable to login to NEM

Having somehow lost it, I downloaded the NEM wallet again on my iPad and then opened it for the first time. It wants me to create a new account and provides no mechanism for me to login to an existing account. What can I do to access mt XEM tokens.

You need to have private key (64 length key, QR account or wlt file) from your old wallet to be able to restore wallet.

Using the private key, I have accessed my NEM account. Unfortunately my XEM coins are missing as it shows a zero balance. Is there anything that I can do to recover my lost assets.

First import your private key.
For desktop walelt:
Sign up -> Private key wallet -> Choose mainnet, new wallet name, new password and provide private key from backup.
If you provided correct private key your address will be same as before. If you made a mistake address will be different and empty.

I simply don’t understand. I sign up for a new Private Key Wallet… I choose main net, create a new wallet name and password … and then type in the 64 character private key …. the address is not as it was before. How can that happen? I have tried to do this repeatedly and am now so confused that I don’t know which way is up. What is the mistake that I made? Did I type even one character wrong of the 64 character private key, would that give a new and incorrect address?

Yes. If you use only one different sign it will generate complete different address.
So please verify each sign of your private key. it should have only numbers 0-9 and letters a-f.