You can translate it by:

“Violet told Clementine to jump […]”

• “Violet quand elle a dit à Clementine de sauter […]”, this example is not really correct regarding to the next part of the sentence. A correct transformation is: “Alors que Violet a dit à Clementine de sauter, Clementine avait déjà perdu sa jambe”.
• “Violet disant à Clementine de […]” doesn’t make any sense with the next part of the sentence too. The word “disant” tells us that she is doing an action. You could say for example: “Violet disant à Clementine de sauter du pont, était en train de rire”. This form is really close to “Violet saying to Clementine […]”

Your analysis is more or less correct but, though trivial and “twitteresque”, the sentence must be analyzed as a whole.

Removing the first interjection “mdr” which means “lol”, the sentence is structured as “violet qui a dit [something] , ça me tue de rire”. In this sentence, “ça” is a pronoun that refers to all the beginning, which means “violet qui a dit [something] ” can be analyzed as the subject of “me tue de rire”. The situation of violet saying something to Clementine is what makes the writer laugh to death.

As for a suitable translation, yes, I suppose you could say :

Lol, Violet telling Clementine to jump off the bridge while she really lost her leg, this makes me laugh to death

Please note, as you mention “Violet quand elle a dit à Clémentine…” that this would focus on Violet, not on her saying something. For example, « Violet, quand elle a dit à Clémentine… , a été vraiment méchante » (« Violet, when she told Clémentine to…, was really nasty »)