My wife had an acciddent with my car I am the insured hitting the city council fence Will my insurance company take care of it?

If your partner has an accident in a car she is not insured to drive it is highly unlikely the insurance company will pay for any damages, in many countries it is also illegal to drive without insurance and punishments for such serious offence can sometimes include prison time. In the UK it would result in a £1000 fine at the very least and probably a 3 month driving ban.