If you bought LW from the official site, there's no password needed
The only official site is Lime Wire » Official Website & Free Download, BitTorrent «

If you did buy from the official site (payment to LimewireLLC) within the specified timeframe (6 months or 12 months, depending if you bought Pro or extended Pro), go here & try all emails that you might've used.
Lime Wire » LimeWire PRO Lookup

If you didn't buy from the official site, or if your Pro's out of date, you won't be able to download a free Pro installer.