Sounds like you may have air in the fluid & if you ran the engine with the brake fluid in there, some has likely mixed in with the P/S fluid & it would be a good idea to flush the entire system with new fluid.
Here is how I disconnected the P/S low pressure return line at the cooler & flushed my P/S system with SeaFoam Trans Tune, to clean it up inside & get all of the old fluid out, before I installed an inline fluid filter, in an effort to cure some other problems was having. You could use this proceedure to flush your pump, P/S rack, cooler & lines, in order to remove any remaining brake fluid that might cause mischief down the road.
Power Steering Fluid Filter
Disabling the fuel pump so the engine won't run & having the front end raised, so the wheels can be turned lock to lock & a helper to bump the starter & turn the steering wheel while I poured in new fluid, made it an easy task & I got a good flush on the system without getting air in.
After servicing the system, having the front raised & idling the engine also allows for easier air bleeding, as we can cycle the steering wheel to bump the stop for not more than two seconds, so as not to damage the pump & thus help chase enrapped air back to the pump reservoir, so the bubbles in the fluid can pop.
On the flush, if you didn't keep the pump fluid reservoir topped up while the starter was bumped & let it get low & have the pump whip tiny air bubbles into the fluid, it'll take a while to work them all out, without a vacuum pump hooked up in place of the reservoir cap to help pop the foam bubbles. So you'll have to let it sit a while, run the engine again & slowly turn the steering wheel lock to lock to chase more foam back to the reservoir & shut the engine off & wait for the bubbles to pop.
If the fluid has foamed really bad & you continue to run the engine & cycle the steering wheel lock to lock, you'll just suck in & cyle more aerated fluid through the system in a vicious never ending circle!!!!!
Wear safety glasses when doing this, & keep the pump reservoir cap on when running the engine & cycling the steering wheel to bleed the system, as those air bubbles being pumped back to the pump reservoir are under pressure, so if they're big, can pop & give you & the engine compartment a facial!!!!! Don't ask how I found this out when doing my first P/S sysyem flush!!!!! lol At least I had my safety glasses on!!!!
Let us know how it goes.