iPad lag after updating iPadOS 15.5.1

If you're talking about the official iPadOS 15.5 (not 15.5.1), you have two basic troubleshooting steps to perform:

A "forced" restart of your iPad:

• On an iPad with a Home button: Press and hold the top (power) button and the Home button at the same time. When the Apple logo appears, release both buttons.
• On other iPad models: Press and quickly release the volume up button, press and quickly release the volume down button, then press and hold the top (power) button. Only when the Apple logo appears (it might take 15-30 seconds), release that button (ignore all other screens that might appear, the one with the "slider" to turn the device off included).

If the issue persists, something could have gone wrong during the latest update procedure.

I would try to install a fresh copy of iOS/iPadOS. To do this, create a full backup of your iPhone/iPad (1), then restore it to factory settings (you need a computer to perform this step) and, in the end, set it up as new (2).

When done, restore data from your previously made backup (3).

Follow the instructions provided in these support articles:

(1) How to back up your iPhone, iPad, and iPod touch – Apple Support

(2) Restore your iPhone, iPad, or iPod to factory settings – Apple Support

(3) Restore your iPhone, iPad, or iPod touch from a backup

If the issue still persists, contact Apple Support directly using this link:--> Get Support or, even better, make an appointment (*) and have your device inspected at an Apple Store or AASP (Apple Authorized Service Provider) among this list:--> Apple – Location Selector. After checking it out, they'll give you all the available options.

Regards

Giulio

(*) You can do it:

• Online:--> Genius Bar Reservation and Apple Support Options - Apple;
• Contacting a local Apple Store or AASP among the list provided above;
• Using the "Apple Store" app, available and downloadable from AppStore.

Start here:

If your iPhone, iPad, or iPod touch won’t connect to a Wi-Fi network - Apple Support

Internet connection difficulties are usually an indication that your device does not have a routable IP Address - your IP Address usually being assigned by DHCP. When this occurs, your iPad may only have a self-assigned link-local IP address that begins 169.254.x.x.

If you have a VPN App or profiles installed, or a security App such as Norton, these may be interfering with DHCP and preventing your device from obtaining a valid IP Address for your iPhone hotspot - or the WiFi network to which you are attempting to connect. The resolution is usually simple; if installed, you may only need to temporarily disable or remove the Security App and/or VPN.

DHCP is frequently used to automatically obtain an IP Address. Until your iPad has a valid IP Address to connect to your network, you will not have an internet connection. Disabling or removing VPN and/or the offending security App should allow DHCP to operate normally - allowing connection to your network and the internet.

If this resolves your issue, the VPN and/or security App can be reinstated after the connectivity problem has been resolved.

A forced-restart may clear any residual issues:

• On an iPad with a Home button: Press and hold the top button and the Home button at the same time. When the Apple logo appears, release both buttons.
• On other iPad models: Press and quickly release the volume up button, press and quickly release the volume down button, then press and hold the top button. When the Apple logo appears, release the button.