Hi and well done on looking for help.
As you say, words aren’t enough, sounds like your wife will have heard them all before anyway, I know my family had.
It reeally is actios that count. The actions that will help you stop gambling are the same actions that might help rebuild that trust. Actions like posting here regularly, maybe getting to your local Gamblers Anonymous meeting if one is near. Deleting everythign on your phone is a positive move; a better one would be to ask your usual sites to ban you and then install a blocker to prevent you signing up at others. Accountability and limiting your easy access to funds with which to gamvel also helps. The accountability will mean you can’t gambvel secretly, your wife will both see you aren’t gambling, which can help with the trust, and also have the peace of mind that you aren’t out there digging another hole.
Show her this site as well – she can get her own support in the Friends and Family Forum.
Keep posting and let us know what positve actions you are taking.