She isn't insane and she's very far from dumb actually. She's a psychopath. Psychopaths know the difference between right and wrong, they just don't think the rules of right and wrong should apply to them. They're superior beings.
It's because of that feeling of superiority that they may appear to do what some would consider dumb. Lawyering up may have made her look like she had something to hide from investigators. She needed to cozy up to them and learn how much they knew. She believed she could convince them of her innocence. It isn't stupidity, it's supreme egocentricity. There is no one as smart as Jodi.
The letter...her behaviors are indeed abnormal because SHE is abnormal. And yes, depraved without a doubt. The reason her actions and words seem so odd is because she is literally having to fake such emotion. Some will tell you psychopaths don't feel but I disagree. I think they constantly feel varying shades of anger. Every other emotion is felt but at a much more shallow level than 'normal' folks. Let's say someone you love dearly dies - you hurt, you grieve, and you miss them. A psychopath will miss a loved one too - but it will be at a much more superficial level - like missing what that person provided to them. Travis made Jodi look better to others and feel better about herself. There's a lot to miss for her but it will never be the same as what those who were capable of truly loving Travis will feel. A story I can offer up: a couple weeks after my mother passed away, my ex flipped out on me, telling me how selfish and cold I was for not understanding HIS loss. Um, MY mom, dude. But that's a good example - it isn't how they're affected but how it affects them, if that makes sense. He was upset that my grief took the focus off him and what he wanted - which was the sympathy and attention others were bestowing on me. Years before, he also became irate I wouldn't attend a work dinner the same day I got out of the hospital after a C-section.
He told me I made him look bad for not going with him and any 'good' woman would. I didn't go so he retaliated by leaving me for a week with a newborn and a toddler. That's what marriage to someone like Jodi looks like.
So because psychopaths don't feel with the depth and breadth of someone non-personality disordered, they're forced to mimic emotions instead. They know what remorse (or love, sympathy, etc.) SHOULD look like but because they can't really feel it, it's difficult to come across as sincere, especially when the psychopath is in the spotlight. The worst thing she could have done, for self-preservation, was write any letter but psychopaths always know better. She believed she could convince those who loved Travis of her innocence, trauma, wishing she could have saved him, etc. No one is as smart as Jodi.
JMO and FWIW