I just don't see the financial argument for not sacking him.
Let's assume that LvG is on £5m a year. Sacking him in December would have cost the club £7.5m, given he had a year and a half left on his contract. Every league position is worth £1.23m in prize money from the PL. Reaching the group stage of the Europa League awards €2.4m while reaching the group stage of the Champions League awards an automatic €12m. Assuming that we're going to get fifth under LvG, hiring another manager who would take us into fourth spot would a) net us £1.23m + €9.6m (£7.3m) = £8.53m, and b) give us a manager that we could give time to assess the squad ahead of the transfer window.
At this rate, it's more likely that we'll drop below fifth than reach fourth under LvG, so the financial gain could be even bigger. Of course, the new manager's going to be on a high wage as well, but that could be offset by any success further than that, for example reaching the latter stages of the CL, and getting €1.5m for every group stage win in the process of progressing through the group
http://www.totalsportek.com/football/premier-league-prize-money-table-2015/ PL prize money
http://www.totalsportek.com/money/uefa-champions-league-prize-money/ CL prize money
http://www.totalsportek.com/money/europa-league-prize-money/ EL prize money