I have a Hoover steam vac and with the Hoover, there's a plastic piece connecting the vacuum to the reservoir which you pivot out of the way to attach the upholstery handheld attachment. If I pivot this away and not attach the handheld attachment, the machine will spray out the cleaning solution but not suck it up allowing me to allow the cleaning solution to soak in before sucking it up. Remember the urine has had a chance to soak in so just spraying out a cleaning solution and immediately sucking it up isn't going to get all of it.
Another trick I have is to load the machine up with hydrogen peroxide, basically cleaning the carpet with oxygen as hydrogen peroxide reacts with organic compounds. You need to test a small section as hydrogen peroxide has a bleaching effect.
Also I would do multiple passes, each with at least twenty minutes for the solution to soak in with each pass. With pet urine, I would do a hydrogen peroxide pass, a soap solution pass, a water rinse pass and a petenzyme solution pass. Note some of the petenzyme will still be in the carpet after you've sucked it up and will continue neutralizing the urine over time.
I have used bleach on certain occasions but I find that Resolve carpet stain remover is more effective but more troublesome as it foams. I would suggest hydrogen peroxide would be better than bleach. You can get some really concentrated hydrogen peroxide at the pharmacy (35%) if you want a very strong bleaching action but I would stay with the 3% or dilute the 35% down a bit.
Often I would use some high efficiency washing machine detergent instead of the carpet shampoo solution. The high efficiency washing machine detergent differs from regular washing machine detergent in that it's more dilute and is formulated not to foam so much, for the deep cleaners, you don't want foam or you'll be constantly rinsing out the waste water reservoir.
Note, these "steam cleaners" are in fact just deep cleaners and do not use steam. The water is usually only hot tap water. The Hoover has a heater but it does not heat the water, it heats the air that it blows down onto the carpet to help dry the carpet. Real steam cleaners (dry vapour cleaners) cost about $3,000 plus and with those you throw down terry cloths to wick up the moisture and the dirt rather than suck it up with a vacuum. You can get cheap versions but they don't last long and you often have to let them cool down before refilling the boiler.