The youngster - signed from Conference North outfit Gainsborough Trinity - has started every game for the Magpies since making his debut in a 2-0 defeat at Peterborough.
With the departure of highly-rated non-contract defender Adam Tann to Leyton Orient this week, Thordarson was delighted at the permanent signing of the 20-year-old.
"We've agreed terms and he has now signed the contract, which we are very, very happy about," the County boss told the official site.
"He has impressed me with his attitude, commitment and desire. He has given everything to earn this contract.
"His performances have been good and we are now looking to see where we can take him from here.
"We know he offers a lot to the team and he has been getting fitter and stronger with every game."
Needham had originally spent time on trial at Meadow Lane in the summer but had to be released due to the Magpies lack of finances.
However, Thordarson swooped for the former Sheffield Wednesday trainee on transfer deadline-day and Needham has already established himself as a first-team regular.
Needham's signing follows that of Nottingham Forest's Eugene Dadi, who also signed until the end of the season yesterday.
N.B. Picture courtesy of the Official Website
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