I thought it menat Elizabeth made absolutely sure that the horses could not be spared before letting the issue go. The exchange reads:

"I had much rather go in the coach."

"But, my dear, your father cannot spare the horses, I am sure. They are wanted in the farm, Mr. Bennett, are not they?"

"They are wanted in the farm much oftener than I can get them."

(From this remark, I gather that Mr Bennet often makes exception even when he feels he really needs the horses - so if Jane asserted her wish to use them she would most probably have pervailed?)

"But if you have got them to-day," said Elizabeth, "my mother's purpose will be answered.

(To me, Elizabeth was saying: "But if you insist on using the horses today, you would give Mother her way - and you don't like doing that, remember?" - almost as if she is coaxing Mr Bennet into letting Jane go in the carraige)

She did at last extort from her father an acknowledgment that the horses were engaged:

i.e. she made sure there was no possibility the horses could not be spared...