I thought I should update this thread with how my story ended - United did reimburse me for the full amount of my laptop's purchase price! In retrospect, I am somewhat surprised because I didn't 'complain' or embellish or threaten them with bad reviews or discontinued service - all I did was explain my story truthfully. Here's how it went:

I tweeted a short version of my story to United and asked who I should contact.
They DM'd me with a 'social media' customer service email.
I emailed them with the full story (pretty much my first post).
They forwarded it to a customer service representative.
The CSR emailed me and asked me to send all my information and an original receipt for my laptop.
I sent off all the required info via certified mail.
After ~2 weeks of not hearing anything, I emailed the CSR, and got a response from a different person, who asked me for all my information again.
I provided all that info again, but informed them that I had already done this.
About a week later, I heard from the second CSR who said he was transferring my case to the insurance company and that they may ask me to send them my computer to verify the coffee spill damage.
I told them that was fine.
The next day I heard from the insurance agent with an offer for a settlement for the full price of my laptop.
I accepted, signed, notarized the document and faxed back.
I waited to receive the check for ~2 weeks, and then contacted the insurance agent that I had not yet received the check.
He said there must have been a mix up and the check had not been issued, but he would remedy the situation.
I received the check 3-4 days later.

All in all, I didn't really do much besides tell them my story and then just go along with the procedures they had in place. It was all through twitter and email (mostly email). I was 100% willing to send them my damaged laptop as evidence, but they ended up not asking me to do that. It probably took 2 months total from the time of damage to the time I received the settlement check. I am so grateful that United agreed to a settlement for the full price of my original laptop, because otherwise I would have been without a computer for a long time until I could save up for a new one. So, a pretty happy ending here.