For anyone interested in replacing the WiFi module, link to ESP-07S data sheet below:

https://www.rfsolutions.co.uk/downloads/1494602250ESP-07S_Datasheet.pdf

Depending upon your location and your preferred supplier, the module will cost between £3.00 and <£2.00, e.g. http://www.electrodragon.com/product/esp-07s-esp8266-serial-wifi-module/

You will also need a IPEX/u.FL to SMA adaptor coax lead, similar to:

http://www.ebay.co.uk/itm/IPX-u-fl-to-SMA-Jack-Female-Bulkhead-Pigtail-Cable-Mini-PCI-RF-Assemblies-15cm-/282614159063?hash=item41cd1dd6d7:

and a

2.4GHz antenna, similar to:

https://www.wifi-antennas.co.uk/2db-sma-male-omni-directional-antenna-2-4ghz.html

I would guess that most hobbyists will have a spare 2.4GHz antenna somewhere in the house (i.e. supplied with that old WiFi router you previously upgraded) however if you do need to purchase, make sure you get one with a SMA to match the coax lead (or if you already have a RP-SMA antenna, make sure you purchase a Coax with a RP-SMA connector to suit! - you cannot mix RP-SMA and SMA connectors.

Couple of observations after looking at the OP's photo.

Firstly, if proper wiring convention was used, then the + and - wires are going to the wrong terminals, of course if the OP had connected the Black (-) wire to the positive terminal of the PSU, then no harm done

Secondly, and I fully accept this may have nothing to do with the overheated terminal block - but noticed that the wiring of the stepper motor connectors do not match the phasing colour indicated on the Duet wiring diagram - I am a novice at 3D printing and NEMA/stepper motor wiring, so hold my hands up as I do not know how important it is to have the correct sequence to maintain phasing - is this important?