A scientist is resetting the clock on his microwave one day as he considers the hopelessness of keeping the time exactly right. Not for the reasons we worry about like power outages and daylight savings times, but because in the back of his head, he knows that Earth's movements through space are not properly standardized for good time keeping system.
This makes him unreasonably mad, so he decides that the best way to correct for this aberration is to simply alter the spin and orbital period of the earth so that a day is exactly 86400 seconds and a year is exactly 365 days based on caesium frequencies so that he never has to worry about converting units again.
Although the historical definition of the unit (seconds) was based on this division of the Earth's rotation cycle, the formal definition in the International System of Units (SI) is a much steadier timekeeper: it is defined by taking the fixed numerical value of the caesium frequency ∆νCs, the unperturbed ground-state hyperfine transition frequency of the caesium 133 atom, to be 9192631770 when expressed in the unit Hz, which is equal to s−1.1 Because the Earth's rotation varies and is also slowing ever so slightly, a leap second is periodically added to clock time[nb 1] to keep clocks in sync with Earth's rotation. ~ https://en.wikipedia.org/wiki/Second
So, our mad scientist devises a two step plan to unify metric and traditional time once and for all! The first stage it to use a series of powerful explosions to speed up/ slow down the Earth's movements to make days and seconds the right lengths, the second it to install propulsion systems on the Earth to keep it moving at these speeds indefinitely.
The Question: How much force (and in what directions) does the scientist need to exert on the Earth to achieve his goals?
Bonus points if new seconds actually = scientific seconds when rounded to the level of a double floating-point number, but I would be surprised if anyone could actually find measurements on the Earth accurate enough to do this, so no pressure.
Caveats based on Comments:
How long is indefinitely?
If additional thrust needs to be added over time, the scientist trusts future generations (assuming he hasn't killed everyone) to continue his work. His initial thruster just needs to be strong enough to make sure that he doesn't start seeing desync start creeping back in before his own end-of-life. If thrust needed decreases over time, assume his thruster can throttle down to compensate.
the explosions and "propulsion" would probably devastate the biosphere at worst it might even generate enough energy to melt the crust and boil the oceans to a significant extent.
Devastation does not necessarily need to be addressed for purposes of this question unless it involves there not being an Earth left to have a day/night cycle.