Latitude 5480, WD15 dock, Dual monitors

Let's backup. What are the two specific monitor models? Link to the specific adapter you are using to try to connect the second monitor to the WD15. Read through the WD15 article, "How to Use and Troubleshoot the Dell Dock WD15".

I have the same issue:  A Latitude 5480 connected to a Dell Dock WD15, with two Dell monitors connected: an E2210 and P2210.  The connection to the E2210 is via a standard VGA cable and it works. The connection to the P2210 is an HDMI connector on the monitor end, and a small USB-type connector on the Dell Dock with a funny symbol; it's between the VGA and HDMI inputs.  The connection to the P2210 does not work.

Ok, I checked Google Images.  The output between VGA and HDMI is Mini-DisplayPort, and my guess is that the DisplayPort output on the WD15 doesn't support "dual mode DisplayPort", which means the more common "passive" DisplayPort-to-HDMI adapters won't work, since they rely on the source to fall back to HDMI/DVI signalling.  If you needed to convert that output to HDMI, you would either need an "active" adapter that can convert a native DisplayPort signal to HDMI inside the adapter itself, which is less common and more expensive, OR given that your E2210 is using VGA (and assuming you're ok with the inferior image quality of analog signals) and your P2210 has an HDMI input, then why don't you just get a regular HDMI cable to connect the P2210 to the WD15's HDMI output, no need for any adapter?

A "small USB-type connector on the Dell Dock with a funny symbol"??  Can you perhaps take a look at the WD15's manual on support.dell.com to find out what port you're talking about, or at least post a photo of the port you're using?