Hello dmorton,
Welcome to the HP Forums!
I understand you would like to connect your Envy 5531 using an iPhone. In order to achieve a solid wireless connection, you want to connect with a router. I can't guarantee a hotspot can work or have a solid connection.
Now, I am going to provide a HP document on Printing from an Apple Mobile Device. There is three different ways you can print from your iPhone. Another suggestion I will provide is using Wireless Direct, which is a added feature on your printer. Please follow this HP document on Wireless printing center.
This document explains information and setting up Wireless Direct:
HP wireless direct allows your Wi-Fi capable devices (Smart Phones) to make a wireless network connection directly to your printer without using a wireless router or access point. You connect to your printer’s HP wireless direct signal the same way you connect your Wi-Fi capable device to a new wireless network or hotspot. (This information was stated here.)
Please post your results, as I will be looking forward to hearing from you.