Hi Fly.
Okay, it's probably best to do three or four charts to give you a good idea of how to go about it, but here goes with your chart.
Now, the idea is, to find the planet that has the greatest amount of essential and accidental dignity, but the accidental dignity is important because the planet's essential dignity is it's latent strength, but it's accidental dignity, is it's ability to act upon that strength.
I am presuming that you already know the difference between essential and accidental dignity?
Just in case, a planets essential dignity, is how strong it is according to the placement in the zodiac, i.e. Mars is very strong in Aries, but very weak in Libra. There are five essential dignities, and to seek the LoG you need to know them all, and not just the domicile and detriment of modern astrology. The essential dignities are, and the rule of thumb, rough 'scoring' for the strength of that essential dignity goes as follows.
Domicile +5
Exaltation +4
Triplicity +3
Term +2
Face +1
Peregrine 0 (Lilly says '-5', which makes no sense because that's detriment, some say '-3' which again makes no sense because peregrine has no dignity, I learned '0', and I stick to it).
A peregrine planet is one that is void of any dignity, it can be influenced for better or worse by receptions and aspects with other planets.
There are debilities as well, such as your Moon in fall, but you only look for dignities for this purpose. In any case the essential debilities are:
Detriment - 5
Fall - 4
Accidental dignity on the other hand accords to a planets position by house, i.e. first house very strong, twelfth very weak. Retrograde motion, combustion, cazimi etc. are also accidental dignities, but the order of the 'power to act' of the houses, is as follows:
House 1
House 10
House 7
House 4
House 11
House 5
House 9
House 3
House 2
House 8
House 6
House 12
Remember that this is a rough rule of thumb and is intended as a guideline only, to be used with your imaginative license in interpretation and common sense, it isn't a mathematical formula, just a guideline.
In general, obviously angular houses are strong and cadent weak, but we can see from the sliding scale which of the angular houses are strongest and which of the cadent weakest.
A person may have a planet in it's triplicity which is a moderate essential dignity, and a planet in it's domicile which is strong, but if the former planet is in the first house, the strongest house, then it has a great amount of power to act upon the moderate strength that it has. Let's say that the domicile planet is in the twelfth house, it is essentially stronger, but so weak for it's accdiental placement in the gaol of the twelfth house, that despite it's strength, it is pretty powerless to act upon it.
Hopefully, you follow me so far.
Now, when seeking to identify the Lord of the Geniture, have a look and see which are the most essentially dignified planets in your chart.
Now, you have no planets in domicile, nor in exaltation, so the next dignity down is triplicity (+3), which is moderately strong, and we see that you have Saturn in it's triplicity, as it is in Gemini, and Saturn is in his triplicity when in air signs between dawn and dusk.
This does not mean that we have found the LoG yet however, we must confirm this. Your Moon in in it's fall, and a very troublesome Moon it will be, because as it is in your ascendant, the strongest house, it has full power to act upon it's fallen state (I have a troublesome Moon too), so the Moon is definately not lord of the chart.
This leaves Jupiter, now if Jupiter happened to be in it's term (it isn't), even though this is a more minor dignity than the triplicity of Saturn, then because Jupiter would have more accidental strength by virtue of it's house placement, it would be able to make the most of it's strength even more so than Jupiter.
So in your chart, like my own, Saturn is the Lord of the Geniture, your guiding light in life, your guardian angel, and the planet whose example you should follow to lead you to your destiny, and Saturn's configuration in your chart will give you clues to this.
Saturn is in the seventh and rules the third. What does this tell you about your self and your purpose?
In traditional astrology, our lives are fated and our fate is written in the horoscope, but we can utilise freewill via the lord of our horoscopes, and this leads us to our destiny, for some it will be more difficult than others, but the LoG is the key to all our unfulfilled potentials, describing our purpose and mission in this life.
It can also be an eye-opener to look for the anti-lord as well, which will always try and lead us astray.
Your weakest most troublesome planet is the Moon, and being in the first has full power to act upon it's malefic potential. The Moon also disposits Saturn, so yes, a difficult Moon indeed.
The Moon in your chart is also opposite the house of it's traditional rulership, the Moon doesn't like to be in the first at all.
I hope you've got some idea of how to work the LoG, or the strongest planet in the horoscope.
In your chart it is quite easy, because it is one of those charts in which the planet simply with the most essential strength is the LoG, because it has the accidental strength to back it up. There are many charts however, in which a planet that does not have as much essential dignity as another is the lord of the chart, by virtue of house placement.
I will find such an example.
Hopefully, you understood me, but I'd be happy to explain again with a different example.