Regret or repent (נָחַם) appears three times regarding Saul's being made king:
The word of the LORD came to Samuel: "I regret that I made Saul king, for he has turned away from Me and has not carried out MY commands." Samuel was distressed and he entreated the LORD all night long. (1 Samuel 15:11) [NJPS]
"...Moreover, the Glory of Israel does not deceive or change His mind, for He is not human that He should change His mind." (1 Samuel 15:29)
Samuel never saw Saul again to the day of his death. But Samuel grieved over Saul, because the LORD regretted. (1 Samuel 15:35)
In the first, it is the LORD who uses the term; in the second it is Samuel; in the third it is used by the narrator to restate what the LORD said to Samuel, the first time. (It may also be a later reflection of Samuel.)
First, the LORD says he regrets that He made Saul king. When Samuel hears this he is distressed and entreats the LORD all night long. "Entreat" is זָעַק which means to cry out. We are not told what Samuel said, but earlier, when the Philistines were coming to attack Israel, the exact phrase was used:
Thereupon Samuel took a suckling lamb and sacrificed it as a whole burnt offering to the LORD; and Samuel cried out to the LORD in behalf of Israel, and the LORD responded to him. (1 Samuel 7:9)
If Samuel's reaction to hearing the LORD regretted making Saul king was to plead with the LORD to change His mind and allow Saul to remain as king, then the LORD's response to Samuel's plea is what Samuel states in 15:29 and there is no contradiction. In other words, Samuel is giving Saul the LORD's answer to his (Samuel's) plea to change His (the LORD's) mind:
1 Samuel 15:11 The LORD tells Samuel He regrets making Saul king
Samuel responds by entreating the LORD [to change His mind]
1 Samuel 15:29 Samuel tells Saul the LORD is not like humans; He does not change His mind
[Samuel is giving Saul the LORD's answer to his plea in 15:11]
1 Samuel 15:35 The narrator reprises Samuel's initial response when the LORD told Him
He regretted making Saul king