This is a scrappy version of chai tea I made in an office microwave with things available in the pantry - a (black) tea bag and hot water and milk from the coffee machine. It did not come anywhere close to the real thing. How do I improve my method?
"Chai tea" usually refers to masala chai, which would be brewed with a lot of spices. If you're looking for a quick and easy way to get closer to that, you can add cardamom, ginger, cloves, cinnamon, and/or star anise to your tea (or buy a box of commercial "chai tea" teabags). (Personally I would say the ginger and the cardamom make the largest difference, especially if you add fresh ginger).
If you're just looking for black tea brewed in a specific way (i.e., simmered) then I think you might need to give up on your microwave/electric kettle setup. You want a consistent heat source so that you can keep the water at a boiling temperature and simmer it with the black tea. I'm guessing you were trying to use the microwave to simulate that, but in my experience microwaved tea is uniformly terrible.
If you want fancy tea at work, you could make it at home the night before, bring your cold milk tea in a thermos, and then add some boiling water to warm it up.