The very start of mishnayos maseches Midos mentions that the l'viyim guarded the bes hamikdash in various locations, and that, if the captain of the guards caught a guard asleep on the job, he'd hit him with a stick. The Rambam's commentary to the start of chapter 5 of mishnayos maseches Sh'kalim says that this punishment was carried out with a strap. How do we explain the contradiction between the mishnayos, or why or on what basis does the Rambam explain Sh'kalim as applying to the sleeping-guard case, which Midos clearly says featured use of a stick?
It can't possibly be referring to the supervisor's a walking stick, as you may not bring a walking stick into the Har haBayit, as we learn in Brachot (Mishna 9:5):
לֹא יִכָּנֵס לְהַר הַבַּיִת בְּמַקְלוֹ
Add a third Mishna and you have a solution.
Makos Ch. 3:12-13 describes that when giving Malkus (the 39 lashes), the instrument used had a handle 1 Tefach long, from which straps were attached.
So we can reconcile your 2 Mishnayos and say that the person in charge of the Levi'im had a stick with straps attached - like Bet Din used.
Midos 1:2: He hit him with his stick, to which straps were attached.
Shekalim 5:1 He whipped him with the straps, connected to his stick.
The above can be proven by the words of the כסף משנה in הלכות כלי המקדש והעובדים בו who says that the instrument used was the same they used for Malkus:
שלוקה מן הממונה בפקיע. אמינא מאי פקיע? נגדא! ופירש''י נגדא: רצועות מלקות
Actually it's a Gemara in Yoma 23a that says:
ולא עוד אלא שלוקה מן הממונה בפקיע אמינא מאי פקיע נגדא
To which Rashi comments: נגדא. רצועות מלקות.
For that matter, if you look in the Rambam in הלכות כלי המקדש והעובדים בו - פרק שביעי he says:
ד: זֶה שֶׁעַל הַשּׁוֹמְרִים הוּא אִישׁ הַר הַבַּיִת שֶׁמְּסַבֵּב עַל הַלְוִיִּם בְּכָל לַיְלָה וְכָל מִי שֶׁיָּשֵׁן עַל מִשְׁמָרוֹ מַלְקֶה אוֹתוֹ בְּמַקְלוֹ וְשׂוֹרֵף אֶת כְּסוּתוֹ:
So the Rambam himself says it's a strap (in Pirush MaMishnayos) and a stick (in the Yad).