I think you have to give the employee something that states how their redundancy pay was calculated - so based on Age, Length of Service, Max weekly entitlement.
I would expect that the JobCenter will want to see some kind of letter from the employer stating that the employee was made redundant, not sacked - so that your ex-employee can start to claim Contribution Based Jobseekers Allowance. I recall taking a very basic letter from my employer to the JobCenter when I signed-on following being made redundant.
Not sure how to resolve the payslip problem. I don't think redundancy pay is taxed. Not sure about NICs/Employers NICs though. I take it HMRC's Employers Helpline were not very helpful regarding that.
HMRC: Redundancy Factsheet may be of help. It says employee does not pay Tax or NICs on redundancy payment, subject to it being less than £30,000. If you have already give employee P45, then the Factsheet says that you should send a letter to HMRC stating the amount of redundancy pay paid, date and any tax deducted - a copy of this letter should be given to Employee.
Don't know if any of that is of any help, or not.