The German has been appointed to the Tynecastle post on a two-and-a-half-year deal but his former employers are threatening legal action over cash they say they are owed, despite relieving the 45-year-old from his Oakwell duties in October.
But Budge has vowed to fight all attempts to take Hearts to court after finally securing her number one target to replace Craig Levein following a prolonged six-week hunt.
The Gorgie chairwoman said: "I'm satisfied that at this moment in time we've nothing to suggest there is any compensation due and on that basis we're carrying on.
"I was as frustrated as everybody else that the process was taking some time.
"We wanted to do it properly but we've got here and I'm sure it was worth the wait.
"Any complication that arises in a process is going to create a delay a bit of a delay, so without question (Barnsley's intervention) did delay the process.
"Was I ever in any doubt? It would be wrong to say I didn't ever have any. There were times where I thought this is difficult.
"But we were persistent, I wanted it to happen and so we kept going.
"For sure, (the ball is in Barnsley's court). I'm sure you've all read the statement Barnsley put out, the statement that I've put out and the statement that Daniel has put out.
"As far as I'm concerned, we're getting on with business."
Stendel was dismissed by Barnsley two months ago following a run of 10 games without a win but the Tykes claimed this week they had yet to agree a severance deal with the German, who they also accused of breaching his contract by holding talks with a rival Sky Bet Championship club.
The former Hannover coach refused to discuss those allegations as he held his first press conference as Hearts boss, but insisted his Oakwell contract was ended with "immediate effect" in a statement posted on his official Facebook account.
While Budge will be left to handle that impending legal battle, it is now down to Stendel to rescue Hearts from the threat of relegation.
The chairwoman was impressed by the passion he showed during their negotiations and he will need to fire up a side currently languishing joint bottom of the Ladbrokes Premiership.
But Stendel is feeling optimistic.
He said: "Our first talks were positive and after that the feeling of trust I got here was more important than money and other things.
"I felt the club wanted me, my personality and my ideas of how to play football.
"There is a nice stadium and really good supporters. The reaction after I signed on social media was so positive and I hope I can bring the success back to Hearts.
"We need to change the game on the pitch. I cannot say we'll win the next three games but we can show more confidence, play more in front so that you can see this is the Hearts style.
"We want to excite the fans so that they say, 'OK that was a nice game, next week we'll come back'.
"We want to win more games and when we win more than we lose, we'll step up the table."
Stendel hopes to eventually be joined in Edinburgh by his Barnsley number two Chris Stern, but for now will be assisted by Austin MacPhee – who remains in the running to be named the club's new sporting director – and coaches Jon Daly and Liam Fox.