Wireless adapter won't get IP address

Do you have access to the DHCP Server?
Does the DHCP Server have a limited range of IP leases?
Is the static IP in a different range than what the DHCP Server is issuing?
If you can check the DHCP Server to see if all the leases have been issued. Test for orphans by pinging; then delete/expire the lease if no response; then on the DELL D530 try IPCONFIG /releases, IPCONFIG /renew (or reboot).

Using Wireshark (I recommend the 'PortableApps' version on a USB stick... it's windows-only and starts up a little slower like that, but you don't have to install it on any windows machine where you want to run it), from any computer on the LAN, to monitor DHCP traffic (with a BOOTP 'view' filter applied) should show all the broadcasts of the entire DHCP process. If everything is being passed on broadcast (you *are* using a standard /24 IP range, right?), then use wireshark to check *from* the machine with the WiFi adapter to verify if it's actually seeing the DHCP offer.

DHCP server has many unassigned IP addresses available for assignment.  The static IP address that I plugged in is in a different part of the range than DHCP is using.  I tried ipconfig /release, then /renew.  The wired adapter got a renewed address OK - the wireless adapter, not so much.  It shows 169.254.178.72.  I'm not sure what this address is or where it came from.

I'll look into this Wireshark suggestion.

169.254.x.x /16 is an address in the APIPA range. Network adapters with TCP bindings set to allow automatic configuration will assign themselves an address in that range if no response from a DHCP server is seen.

What IP range is the DHCP server set to use?  (i.e. its Scope.)

Did you watch the process with Wireshark?

You can trigger a new request by disabling the adapter, then enabling it again. The easiest place to do that is by right-clicking on the adapter in Network Connections (ncpa.cpl).

DHCP can assign addresses in the range 192.168.4.69 thru 206.  There are plenty of unassigned addresses available.
Disabling then re-enabling the adapter didn't work - still no addresses for the wireless.

I'm still working with the portable version of Wireshark.  No success in getting that to run yet.  It's a bit of a stretch for me.

Do you thing Winsock might have anything to do with this - or not since the wired connection is working.

I didn't offer a very technical solution and didn't even think about which app was managing the wireless connection when I posted the question, but when the wireless starting working with Windows managing it versus the Dell manager -- case closed.