B.M.
You're actually 'helping' her to lie by the way you are interacting with her. It's developmentally age appropriate for her to be figuring out lying at this age and it's your job to HELP her NOT lie. not to punish her for catching her. You goal is to teach her how to behave.
First of all, you know she has impulse control problems, so eliminate them for her. Take the cord to the TV so she can't turn it on without you.
Next - if you KNOW she did something wrong, don't ask her if she did it. That's game playing. State the fact and then discuss it. "Susie, you were watching TV this morning and that is against the rule. You will not be able to watch TV for the rest of the day". You eliminate the lying all together.
if you KNOW she threw the puzzle at her brother you say "Susie - we do not throw in this house. Apologize to your brother and go play by yourself in your room (or the dining room, or whatever) until you are ready to behave correctly."
Also - her motivation for telling the truth shouldn't be so you don't get mad. YOUR emotions should not be the driving factor in why she behaves. She's not responsible for how you feel. So separate your emotion from her behavior and then deal only with HER behavior. Using the 'mommy will be sad if you...." is only developmentally appropriate for kids who are like 3 when you are teaching them empathy. At this point it's time to focus on behavior.
"Susie - lying is not something we do in this family. You lied about eating the cookies. Therefore, you will not be able to eat cookies for the rest of the week" or whatever.
I disagree that the consequence should not happen if she tells the truth. She is 6. She knows the rules. If the rule is that you ask before turning on the TV, then if she turns on the TV she loses that priviledge for the rest of the day. The consequence for lying about it is that she loses independence. If you can't trust her to be up without turning on the TV then she can't enter the room where the TV is until you are there. 2 different issues, 2 different consequences.
There should not be punishment that is not related to the issue - what does putting pepper on her tongue teach her? It teaches her that you will cause her discomfort if she breaks a rule. Instead - TEACH her what to do and why it's important.
Good luck.