Open your email signature in your device's browser app.
NOTE: Do not open your email signature in your mail app, you must open it in your web browser app.
Tap and hold on some text in your email signature such as the name, then tap Select All. Your email signature will be highlighted, then tap Copy.
Go to your Settings app and tap Mail, then scroll down and select Signature.
Select either All Accounts or Per Account, whatever is preferred, then tap and hold in the field below. Select the Paste option to insert your copied email signature.
You will notice that your email signature isn't quite displaying correctly. Shake your device side to side until a text box opens prompting to undo changes. Tap Undo and you'll see your signature's formatting is corrected and displays as intended.
Close out of your settings and compose a new email. You should see that your email signature is automatically added to the end of your email.