A brace from Josh Dasilva, alongside strikes from Emiliano Marcondes, Bryan Mbeumo and Tariqe Fosu, completed the Griffin Park rout and left the Owls with just one Championship win in 10 outings.
“That was the most embarrassed I’ve ever felt in football. It was inexcusable. I’ve defended the team before but we were hiding today,” said Monk.
“We have responsibilities in terms of what we face on a football pitch but the one thing you need is heart and we didn’t show any of that. I can only apologise to the fans behind the goal because they don’t deserve this.
“Every time we take a step forward we take two steps back. I have said the boys train well and hard and a lot of it you don’t see coming, and then you get that on the pitch.
“There is a leadership problem and I share responsibility for that. I have to lead that team and those players on a daily basis, but I’m clearly not getting a response.
“I understand that but I can only do what I can physically do. I have done all I can and it’s not had the response that’s needed. We were nowhere near it today.
“There’s a culture here that will never ever be successful. We don’t have enough desire, determination and character and if you want to be successful there has to be desire beneath you. There is not enough of that at this football club.
“It’s not an individual thing. It’s the whole team and there are deep rooted issues here for everyone to see. I am tired of saying it.”
Asked about changing systems to combat a rampant Brentford side, Monk added: “The system wouldn’t have mattered because if you don’t play with heart you will be punished.
“We did well until Christmas but we’ve had a huge drop off since then and trying to find a way through it to rediscover that consistency is proving a very difficult thing to do.
“But I’m not a quitter and I’ve had to fight for everything in my career and I will also fight to turn this round until someone tells me otherwise.”
Brentford: Raya, Dalsgaard, Jeanvier, Pinnock, Henry, Marcondes, Norgaard (Baptiste 78), Da Silva (Valencia 84), Mbeumo (Fosu 79), Watkins, Benrahma. Unused substitutes: Dervisoglu, Daniels, Luca Racic, Roerslev Rasmussen.
Sheffield Wednesday: Dawson, Palmer, Iorfa, Lees, Fox, Murphy, Pelupessy, Bannan, Da Cruz (Forestieri 46), Windass (Wickham 46), Fletcher (Harris 65). Unused substitutes: Nuhiu, Wildsmith, Hunt, Urhoghide.
Referee: K Stroud (Hampshire).