I'm sure that was meant as a rhetorical question - but I'll answer it anyway!
The last time we were unbeaten in January was in 1979.
Our league position then was very similar to now, we were 16th in the second division.
We only played 3 games in the month, all at home, and two of them were memorable matches.
The first one was on new year's day, when we beat Oldham 2-0.
We had three debutants that day - Bobby Smith, Dave Buchanan and Gary Lineker. I was so impressed by one of them he instantly became my favourite player. Sadly I hadn't spotted the potential of the future England captain, it was Bobby Smith who caught my eye. I think I only liked him because he had a proper footballers haircut and came from my favourite Scottish team, Hibs. Smith and Buchanan both scored. Lineker had to wait nearly 4 months for his first goal.
The second match of the month was on the 6th, in the cup against Norwich, when we won 3-0.
Bad weather at the time meant that most of the day's matches were called off, so the match of the day cameras were diverted to Filbert Street, where the tent ensured the match could take place - and Keith Weller could score a wonder goal (and his last for Leicester), in his white tights. I was standing at the front of pen 4 at the time and had a perfect view of his mazy run. Larry May and Martin Henderson got the less memorable goals. Norwich were in the first division at the time, so our comprehensive win was a bit of a shock.
The final match was on the 20th, when we drew 1-1 with Blackburn. I can't remember anything about this match. Larry May got the goal.
We only stayed unbeaten that month due to the weather. Our fourth round FA cup match away at Oldham was postponed, and they beat us 3-1 in the rearranged game, which didn't take place until the end of February. Alan Young scored a hat-trick for Oldham, and was signed by Leicester soon afterwards.