Get a bench power supply set for the battery voltage and apply to the voltage input tangs to verify detector works.
If it does it comes down to just a contact problem. (Check the contact on the battery too - or replace with the Alkaline module) Sounds like you don't want to send it to Minelab? Out of warranty?
If it happened to me and out of warranty I would put on an Optivisor to see what I was doing and use an exacto knife or Dremel to scrape away the corrosion on the contacts.
Then I'd perhaps try using a very high temperature setting thin tip soldering iron (order online) to tin the exposed copper on the tangs. (heat sink the part of the tang touching the plastic - or be real fast - so it doesn't melt it) Then clean tang with isopropyl alcohol or flux cleaner.
Perhaps then coat with one of those expensive contact improvers like DeOxit.
A step up would be to electroplate the contacts - check into gold electroplating kits.
A super fix might be to take the contacts out of a charger base to replace the set in the detector box. Might be the same part.