Well, this was a nice distraction, thanks!
First off, what piece, or composer, is this? (I'm just curious.)
Here's what I've got:
I chose this route for a few reasons:
The odd time signatures may seem strange, but it's certainly not written in 4/4. You'll need to have a few oddities and a few changes in an excerpt like this, it's just the facts of life. (Hey, blame the composer!)
Furthermore, the 7/16 (or 7/8, if you choose to double all note values) shows the parallelism between the first two measures. In other words, it makes clear that both hands are using the exact same rhythm in both measures.
I changed time signatures in m. 3 to show parallelism again; notice the E-Bf-A-F line in the upper voice starting at the 2/4. This then repeats starting at the 3/8, just an octave higher. (Note that I'm focusing on the RH here to find the meter; one could adjust a time signature here or there so that the LH is the metric anchor, but I find my way to be easier.)
With that said, the composer threw in another little hiccup; instead of a second 2/4 bar, we need something with 9 sixteenth notes instead of just 8; hence the 3/8 plus 3/16.
The concluding 7/16 serves two functions: it matches the durations of the held notes in the right hand, and it also nicely matches the left hand syncopations from the opening two bars.
Lastly, someone might look at this and be troubled with the fact that every left hand downbeat except the first is a tie from the previous measure. At first glance, this would seem to suggest that I've chosen incorrect time signatures, until you realize that the right hand articulates every downbeat. Thus, no matter how one notates it (unless it's a string of 1/16 measures!), there will be tied notes held over into new downbeats. As I said in Step 3, I chose to use the RH to find the metrical downbeats, so here's our result.
The final two pitches could be notated in a variety of ways; we'd need to hear the next section to know for sure, but a fermata there is a relatively good guess.
EDIT: Things like this are always a little subjective; I'm open to any suggestions!