First of all, spoiler warning, as Visions was just released...

Spoiler: With the help of his X-Wing, Karre cuts a Star Destroying in half, and Am is able to halt him before he goes into hyperspace. A ton of scaling in bound, so hold on tight...
Spoiler:
First up, let's get the width of the area melted. Using the average height of a man (which globally is 171cm, or 1.71m) for Karre until we find something else.
Spoiler: 374 pixels = 1.71m
1 pixel = 1.71m/374 = 0.00457219251m
0.00457219251m X 51 = 0.233181818m
Spoiler: 96 pixels = 0.233181818m
1 pixel = 0.233181818m/96 = 0.00242897727m
0.00242897727m X 104 = 0.252613636m

Next, let's find the dimensions of the Gemini-Star Destroyer. An X-wing is 11.76m across.
Spoiler: 303 pixels = 11.76m
1 pixel = 11.76m/303 = 0.0388118812m
0.0388118812m X 1322 = 51.3093069m
Spoiler: 29 pixels = 51.3093069m
1 pixel = 51.3093069m/29 = 1.76928644m
1.76928644m X 78 = 138.004342m
1.76928644m X 44 =77.8486034 meters
1.76928644m X 241 = 426.398032m
1.76928644m X 801 = 1417.19844m
1.76928644m X 786 = 1390.65914m
1.76928644m X 765 = 1353.50413m
1.76928644m X 790 = 1397.73629m

With that in our hands, let's now find the volume of the section melted. We can see the exact point at which Am managed to block it...
Spoiler: ...so we'll scale up to there.
Spoiler: 27 pixels = 138.004342m
1 pixel = 138.004342m/27 = 5.11127193m
5.11127193m X 979 = 5003.93522m (5.00393522km)
5.11127193m X 4 = 20.4450877m
5.11127193m X 707 = 3613.66925m
5.11127193m X 41 = 209.562149m
5.11127193m X 39 = 199.339605m
5.11127193m X 33 = 168.671974m
5.11127193m X 13 = 66.4465351m
5.11127193m X 128 = 654.242807m
5.11127193m X 410 = 2095.62149m
5.11127193m X 34 = 173.783246m
5.11127193m X 41 = 209.562149m
5.11127193m X 524 = 2678.30649m
5.11127193m X 19 = 97.1141667m
5.11127193m X 249 = 1272.70671m
2095.62149m - 1272.70671m - 209.562149m = 613.352631m

Firstly, we'll go for the total energy to melt the segment of the Star Destroyer cut. Volume of all the sections as rectangles (with a width and height of 0.252613636m and 20.4450877m respectively).

V = lhw
= 3613.66925 X 20.4450877 X 0.252613636
= 18663.5463m^3

V = lhw
= 66.4465351 X 20.4450877 X 0.252613636
= 343.176948m^3

V = lhw
= 654.242807 X 20.4450877 X 0.252613636
= 3378.97302m^3

V = lhw
= 613.352631 X 20.4450877 X 0.252613636
= 3167.7872m^3

We'll add the volumes together, then convert it to cm^3.

V = 18663.5463 + 343.176948 + 3378.97302 + 3167.7872
= 25553.4835m^3
= 25553483500cm^3

We'll go with a low end of aluminum (2836.11219 joules/cm^3) and a high end of steel (7309.87 joules/cm^3).

(Low end)
E = 2836.11219 X 25553483500
= 7.24725461e13 joules
= 17.321354230401528 kilotons

(High end)
E = 7309.87 X 25553483500
= 1.86792642e14 joules
= 44.6445129063097497 kilotons

With that all done, let's find the energy per second.
Spoiler:  Timeframe is 10 seconds and 8 frames.

T = 1s/24
= 41.6666667ms X 8
= 0.333333334s + 10s
= 10.3333333s

(Low end)
E = 7.24725461e13/10.3333333
= 7.01347223e12 joules
= 1.676260093212237 kilotons

(High end)
E = 1.86792642e14/10.3333333
= 1.80767073e13 joules
= 4.3204367351816444 kilotons

That's also the minimum energy for Am stopping Karre's attack. Karre wins out though when R-DUO kicks into hyperdrive, and splits the Star Destroyer in half proper. Let's find the kinetic energy of the split. We'll calculate the armour larger two parts of the armour as two triangular prisms (it's not exact, but given the shape of the Star Destroyer and not taking the bridge into account, it'd be a low end in anycase).

A = hb/2
= 5003.93522 X 1417.19844/2
= 3545784.59m^2

V = bh
= 3545784.59 X 20.4450877
= 72493876.9 X 2
= 144987754m^3

Keeping consistent with what we've been working with above, we'll have a low end of aluminium (2600kg/m^3) and a high end of steel (7850kg/m^3) for our mass (both values here).

(Low end)

M = 144987754 X 2600kg
= 376968160400kg

(High end)

M = 144987754 X 7850kg
= 1138153868900kg

Now for the middle, we'll go with two triangular pyramids and a triangular prism.

A = hb/2
= 1353.50413 X 426.398032/2
= 288565.749m^2

V = 1/3hb
= 1/3 X 288565.749 X 5003.93522
= 481321438 X 2
= 962642876m^3

A = hb/2
= 5003.93522 X 1353.50413/2
= 3386423.49m^2

V = hb
= 3386423.49 + 97.1141667
= 3386520.6m^3

V = 962642876 + 3386520.6
= 966029397m^3

As per usual with vehicle and machine mass calcs, I will be using mass of the light ship mass of container vessel 2700TEU, that being 102.56kg/m^3 (this is likely a huge low end, but it's the best I've got).

M = 966029397 X 102
= 98534998494kg

Adding those to our above values...

(Low end)

M = 376968160400kg + 98534998494kg
= 475503158894kg

(High end)

M = 1138153868900kg + 98534998494kg
= 1236688867394kg

Our penultimate directive is to find the speed.
Spoiler:  Timeframe is 2 seconds and 12 frames.

T = 41.6666667ms X 12
= 0.5s + 2s
= 2.5s

407 pixels = 1397.73629m
1 pixel = 1397.73629m/407 = 3.4342415m
3.4342415m X 122 = 418.977463m

T = 418.977463m/2.5s
= 167.590985m/s

At last we shall have our energy!

(Low end)

KE = (0.5)mv^2
= (0.5) X 475503158894 X 167.590985^2
= 6.67766638e15 joules
= 1.596000568833652045 megatons

(High end)

KE = (0.5)mv^2
= (0.5) X 1236688867394 X 167.590985^2
= 1.73672783e16 joules
= 4.1508791347992355369 megatons

Final Results
Gemini-class Star Destroyer length = 5.004km
Total energy of Karre slicing through Star Destroyer (low end) = >17.321 kilotons
Total energy of Karre slicing through Star Destroyer (high end) = >44.645 kilotons
Energy of Karre slicing Star Destroyer per second (low end) = >1.676 kilotons
Energy of Karre slicing Star Destroyer per second (high end) = >4.320 kilotons
Karre splits the Gemini Star Destroyer in half (low end) = >1.596 megatons
Karre splits the Gemini Star Destroyer in half (high end) = >4.151 megatons

This also involves a lot of low ends for the mass and I couldn't be bothered doing the bridge, so ultimately the final result is going to be even higher than that. It's also worth noting that about a quarter of the total mass of the Gemini Star Destroyer weighs several hundred million tons.