Functional Javascript Practice with AJAX
Move your APPID and API url from your code an into constants. That way, they're readily visible and easily configurable.
Move the creation of your API url into a function, that way it's easily modifiable. In addition, use $.param to construct the query portion of your URL. jQuery encodes the params for you. If you can do ES6, you can simplify using shorthand literal notation and template strings

function getApiUrl(lat, lon){
  return `${API_URL}?${$.param({lat, lon, APPID})}`;
}

Not entirely sure why you need to stringify data.main.temp when the logic comes after it expects a number.
data.main.temp appears to be in Kelvin. It would be better if you name the variable appropriately to show that it is in Kelvin.
Move your temperature calculations into functions, like convertFromKelvinToCelsius and convertKelvinToFahrenheit.
It appears that you only get the data once, but your conversions to Celsius and Fahrenheit happen on click. Why not convert them just once, and on click, just display the pre-calculated values?
In the same spirit, why not render the HTML in advance and just show/hide depending on the click. This avoids you from putting the values in html all the time.
Construct your data first, then do the HTML. Makes it easier to have all your logic up top, and rendering down below. That way, your data is prepared and all in one place.
The &lt;i&gt; part, you're repeating code. Why not determine the class name only and addClass to the &lt;i&gt; once instead of repeating the addClass for each condition.


I'll leave the code to you. Feel free to open up another review for a followup.