I cannot help you much with the current issue, other than advising you as others did, that if indeed he broke it your account, and you have to ask the provider, you will have to go to court for the provider to give data about IP addresses.
I would also ask gmail if they could produce a backup of that deleted email via the court order. I also suspect that you do not even need to prove it was the actual guy hacking your email, and it will be enough creating a reasonable doubt wether your email account was hacked at that time.
As your comments talk about Google, you can edit a history of what devices accessed your account, and it says the make of the device, and the City used.
Go to "My Account"->Device activity & notifications" and under "Recently used devices" select "REVIEW DEVICES". Select the suspect device in case it is there.
For the future, I advise you to activate double factor auth, 2FA, which obliges a device to use a token.
I also ask you, how was that individual able to hack you multiple times? It it possible that in a point in the past he planted some spy software in your computer? Have you answered to emails "Your account is due to be closed, please confirm your username and password?" The details about your account be compromised seem fishy, to say the least. The easiest way for the guy to go about it, would be to create an email very similar to yours, and use 1 true email together with 48 edited emails from that account...
Obviously this is all for problems in the short term. In the long term of things, most providers are only require to save IP addresses and usage by law for 2 years at least.
Beware also that some other emails (2 years +), are harder to locate in Gmail inboxes.
I would also not wait for him delivering the emails to the court; he has a vested interested on only delivering them at the last minute. Ask through the court for a list of IP addresses, emails and time from google exchanged between your address and the address of the guy.
On the producing of proofs,beware of bureaucracy. In my case, they summoned me for "a simple meeting", and the court did not do the paperwork on the Citius database, they refused to give data about it to my lawyer over the phone, you yourself as non-lawyer cannot have a look at your own process, and I had to send my lawyer to another city for browsing the physical files, and only them we find out they had another type of complaints behing the "meeting", and even had arranged witnesses to give a fake account of the facts.
As other said, try to hire a competent legal aid familiar with this field.