A pair of 3-1 wins in the Premier League, over Liverpool and Hull City, helped Shakespeare land Claudio Ranieri's old job until May, with Sevilla looming in the Champions League as his first test since assuming full-time control.
Leicester's players losing faith in title-winning boss Ranieri was reportedly behind the Italian's dismissal following the 2-1 defeat in Sevilla, which leaves the Foxes with an uphill battle to prolong their European adventure.
But Shakespeare, who began a second spell at the King Power Stadium in 2011, appears to have the full backing of his dressing room.
"It's a great appointment, a good fit," midfielder Drinkwater told a news conference. "We don't have much time to turn things around but we could not have got anyone better at this moment.
"As a manager he has taken on board everything and passed what he can on to the players, which is excellent for us.
"He hasn't changed that much and that is a good thing. He needs to take control of more things as manager, which is his duty, but any changes he has made have been positive.
"He helps us enjoy our football. Just do the simple things and hopefully the rest will shine through.
"He was huge last season. He helped Claudio settle in, and was a link between the manager and players. He's done that excellently since I've been working with him.
"On behalf of all the players, we'd like to see him get the job long term. We were all very happy to see him get it short term, I'm sure it would be the same reaction longer term."
Drinkwater concedes that Ranieri's sacking may have been required to snap Leicester out of a slump that had seen the champions fall into the relegation zone.
"I think it's been a long time coming, I think we've waited a long time, for the results, for the performances," he added.
"It might be down to the change, it might be down to the players finally understanding that something needs to happen and we might have forced it.
"It might be the new gaffer, it's hard to put your finger on. But as long as it has changed and we're now getting the results - that's the most important thing for us."