The shortest day has to be in winter because that is when daylight savings occurs and the days are shorter and nights are longer. I'm not sure which exact day is the shortest, but that should narrow your search.
With that said I invite you to Islam, the religion of truth and justice and the only religion God will ever accept from mankind. I invite you to read the Noble Quran with Ibn Kathir's Tafsir interpretationn which is online and free so as to understand what you read in detail. If you accept and do good deeds and worship only Allah, by whose permission the heavens and the Earth exist, then I give you glad tidings of paradise, Allah willing, for your entertainment in the hereafter and a good life in this world.
If you reject, then I am only a warner just like many people have warned before me. So may Allah guide you to the religion of justice so you can taste this mercy too.
And when My servants ask you, [O Muhammad], concerning Me - indeed I am near. I respond to the invocation of the supplicant when he calls upon Me. So let them respond to Me [by obedience] and believe in Me that they may be [rightly] guided. (2:186 - Noble Quran)