They all have problems.
I handle shipping for my company. UPS is the most reliable, but yes, they lose packages, damage things, and have some incompetent people working for them.
DHL is usually as good as UPS, but they have some hidden fees they tack on for international shipments.
Fedex is far behind the other two in service. They lose packages the most often in our experience, and any call to customer service takes hours longer than it should, when you factor in the time spent on hold and the number of calls it takes to get them to do things right.
My advice is to call DHL customer service and explain the situation. Don't settle for "we know we made a mistake and it will be there tomorrow." They failed on their guaranteed delivery date, and they are liable for the first $100 of merchandise shipped, unless you insured it for more.
Make sure you talk to someone who will find out all the details and personally see to it that the situation gets resolved. They have people whose job it is to do that, but the first line of customer service people, as in any company (UPS included), are just there to get you off the phone.