You can get Priority Pass by signing up directly (not recommended) or via the Amex Platinum, Citi Prestige or Chase Sapphire Reserve credit cards if you live in the US. Please refer to the credit card forum for additional info, I cannot opine on which one may be best for your needs (if you want another credit card).

Will you not arrive to JFK before 9-9:30pm (which would give you time in the Senator lounge to shower)? I would recommend against visiting the LX lounge given you have to clear security twice and deal with the Airtrain transfer - probably not great even now that BR participates in PreCheck.

Your DL Platinum Medallion status would not gain access to the KE or AF lounges because you are not traveling on a SkyTeam itinerary.