The 31-year-old spent the first half of the season on loan at Rangers, but made only two appearances and was recalled by the Foxes earlier this month.
The Wales international has played 329 league games for the East Midlands side since his debut in October 2007.
He could feature for Huddersfield in Saturday's home game against promotion-chasing Brentford.
"There's a real alignment with what we're striving to do and Andy's goals for the rest of the season," boss Danny Cowley told the club website.
"We're fighting to maintain our position in the league and Andy is incredibly determined to do well, as he's out of contract soon and fighting to make the Wales squad for the upcoming European Championships."