You still have to mention all trips since becoming a permanent resident, including more than 5 years ago if you've been a PR for more than 5 years. It's just that when they evaluate your eligibility for citizenship, they generally don't care about trips more than 5 years ago, even though you're supposed to list them.Provide a date which is rather quite close to the trip. As jack mentioned, we only have to mention travel dates for 5 years from the date of application!
You need to go to the US embassy on your country and get help there.
If you tried that I doubt they would even let you past security at the front gate. A country's embassies exist primarily to represent that country overseas - not to help people who couldn't be bothered to keep the correct records.You need to go to the US embassy on your country and get help there.