There aren't really rules around "proper" time signatures, so much as practical considerations. "Is there another way of expressing this?" Yes, there is almost certainly a better way.
The first thing to understand is that a time signature doesn't tell you how many notes are in a measure, or even simply how long the measure it is. It also makes implications about meter, the idea that these notes and beats are organized into recurring structures. This idea of meter inherits from the same idea in poetry, in which you organize words by strong or weak syllables and into lines of a certain number of syllables. Consider:
Twinkle twinkle little star
How I wonder what you are
The strong and weak syllables are arranged into trochees—Twinkle twinkle. And there are four such groups in each line. Compare:
Hey diddle diddle
The cat and the fiddle
This time, the pattern is one strong syllable followed by two weak syllables: Hey did-dle did-dle / the (yes, we cheat a little by printing "The" on the second line, but it really belongs with the first metric line). Here the foot is dactyllic; we have syllables grouped in sets of three, and two such groups in a line.
Musical meters work similarly. There are many questions/answers on here already highlighting the fact that 6/8 is really "compound duple" meter, two beats subdivided into three.
Whew! That's a lot of setup to say:
- Your group of notes probably has some internal groupings. You said you counted them out as "1-e-and-a 2-e-and-a"; this suggests that they fall into beats that subdivide into four. Now, do you find the beats grouping themselves in regular ways? Perhaps the first 32 sixteenth notes are actually two measures of four quarter notes? As others have hinted, if you do in fact have some "irregular" measures, chances are that either they're a number of much more mundane measures with a single irregular one appended, or, even if they do group into more exotic subdivisions like 5 or 7, they still form meaningful groups and phrases.
- I can't help questioning the count of 41. For one thing, it's a big number to count to! But secondly—forgive me if this is something you already totally understand—beat and meter make no claims about "how many notes" are in a measure. There might be 16 sixteenth notes, or 4 quarter notes—or a single sixteenth note, a dotted eighth note, a quarter note, and a half rest. If you are counting actual notes, make quite sure that they're all the same length and that there are no rests involved. Your final "10-e-and-a 11-" ends the bar only if we need the next bar to begin immediately. Is there a chance that the phrase really finishes off "10-e-and-a 11[rests: -e-and-a 12-e-and-a]"? Or that the final note is longer than a sixteenth note?
- Choose a time signature on the basis of how the notes fall together into beats and phrases, and
- It's not just about "the notes"; silences and held notes take up time too!
Since the audio link has been posted, a few things are clear: No, my fears that you were confusing "how many notes" with "how many beats" were unfounded. You've got notes of several different durations mixed in there, and yes, you counted through them accurately. (Or, maybe you did? I keep making it two full bars of 4/4 followed by 2 and a half beats, which would be a total of 42 16th notes. But I'm not at all sure I'm keeping up with the count.) Also, yes, this is definitely an irregular-length phrase. It's far from cut-and-dried how you should organize the beats into measures, and yes, you'll have some kind of unusual time signature at some point.
The first grouping that occurs to me is simply to start in 4/4. If we proceeded with this, we could get two full measures in, and then if I'm hearing the remainder correctly, it would just be a solitary measure of 5/8. A case could also be made for some more unusual groupings, but let's look at what musical elements suggest this 4/4 conclusion.
First of all, in the beginning, the ride cymbal is recurring regularly; this leads us strongly to perceive this as "the beat." If those are quarter notes, then everything the drums do in the first 7 beats is just 8th notes. Another thing suggesting 4/4 is the descending series of four tom notes, which would take up the second four-beat measure.
A case could be made for other groupings within those first 7 quarter-note beats—is it actually 3/4? Maybe 5/4? The only difference in that case is where you want the "strong beat" of a downbeat to be felt.
I would take a "subtractive" approach to explaining what's left over; it's as if there "would have been" a longer bar (and maybe another bar beyond that) but it got interrupted. This is where some unusual time signature will come in handy (but I think it's actually a whole number of 8th notes, in which case I'd use that rather than 16ths).