Disclaimer: the response provided below is informational only and should not be relied upon as legal advice. The response provided does not create any attorney-client relationship. You should consult your own attorney to help protect any legal rights you may have.
Has anyone from the Public Storage company or the insurance company acknowledged receiving your claim, or offered to pay for your damages? If they are not being responsive at all, you will need to make an assessment of the dollar amount of your damages and sue them. Keep in mind that there are time limits for suing on any legal claims, and once these time limits (statutes of limitation) pass, you will be time barred.
How long has it been since you sent all your materials? If it's been more than three weeks, send duplicates certified mail. Make sure you can prove these people get every item you mail them. Follow up with phone calls. Where is this outfit located? What is the total dollar amount of your claim? Is it over $7500? If not, why not just sue both the storage facility and Willis in Small Claims Court? But before doing so, give them a fair chance to respond and pay, and let them know that if you do not hear from them by (choose a deadline), you ARE going to sue. Then do it. I have not heard good things about outfits that provide insurance sold by retail level businesses, so I would keep on them and cross my T's when dealing with them. If they even gave you a copy of the insurance policy, read it very carefully and make sure you comply with its terms so they have no excuse for dragging their feet.
For those outfits that DO like to stall and harass a claimant, some favorite tricks are "I didn't get your letter," "Send me that again, I can't find it," "That's not covered." "Your loss estimate or receipt is not signed," and "That's excluded." DO NOT take their word for what is covered. You can read as well as they can. If the policy does not clearly EXclude it, it is covered.
What you might not think about is that if they simply ignore 100 people, 75 of them will give up and accept not being paid. Be one of the 25. Another popular strategy is to ask for one item to document your claim, then sit on it for a month. Then when you call, they ask for another item. And so on. Make them know you mean business. Selling a service and then refusing to provide it sometimes goes by a five letter word: fraud. Also consider telling them you'll report both of them to the Better Business Bureau as well as suing them.
I hope this helps. P.S.: there is no law that says you have to accept the cheapest estimate of the damage or loss. Get several estimates and choose the ones you think are fairest to submit to them.