This seems like a very cool idea for a project. Nice work so far.

General rule of thumb: If you Google it and don't find anything other than posts asking if there is a level editor for a specific game, chances are that, unfortunately, there likely isn't one. Similarly, if a game is hacked barely or not at all, chances are that someone likely hasn't taken their time to program a level editor for it. A game really only ever seems to be hacked once there is a utility to simplify the modifying process, which can only come about when someone takes the time to figure out how the game works to begin with, and then has the ability to program a utility to modify the data, normally visually to help the process.
I don't know how much information you have already found yourself, so maybe this post won't be that helpful for you.
Using the FCEUXD emulator, I found the following information for the "Gyromite (W) [!].nes" ROM.
With this information at hand, it might help to draw out the levels that you want to design, and then add them in by hand through a hex editor. At least, until someone looks into this game and designs a level editor for it. However, I don't see this happening anytime soon, unless this thread peeks their interest and motivation enough to do so.
Best of luck with your project!

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RAM: ROM
$8000-$9FFF: 0x0010-0x200F
$A000-$BFFF: 0x2010-0x400F
$C000-$DFFF: 0x4010-0x600F
$E000-$FFFF: 0x6010-0x800F
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Game A: 40 Phases. If you win Phase 40, it starts over at Phase 1.
Game B: 25 Phases. Starts from R=01 (Phase 1).
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0x33AB-0x33CA (0x20) - Select Mode & Phase loading screen palette
0x33CF-0x33EE (0x20) - Title screen palette
0x3483-0x34A2 (0x20) - Phase 01 palette
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0x3510-0x3672 (0x163) - Select Mode screen graphical tiles to draw
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0x36B6-0x373D (0x88) - Screen graphical tiles to draw at top of Phase screen (1P score, time)
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0x3939-0x3970 (0x38) - Phase loading screen display - border, Phase, Player
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0x3971-0x3AAF (0x13F) - Title screen graphical tiles & palette definitions
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0x3E9C-0x3EC3 (0x28) - Game A - Order of Phase designs
15 - Phase 01
20 - Phase 02
00 - Phase 03
1A - Phase 04
07 - Phase 05
08 - Phase 06
09 - Phase 07
0A - Phase 08
01 - Phase 09
1C - Phase 10
23 - Phase 11
0B - Phase 12
0C - Phase 13
0D - Phase 14
0E - Phase 15
0F - Phase 16
10 - Phase 17
11 - Phase 18
12 - Phase 19
13 - Phase 20
14 - Phase 21
02 - Phase 22
03 - Phase 23
17 - Phase 24
18 - Phase 25
19 - Phase 26
04 - Phase 27
1B - Phase 28
06 - Phase 29
1D - Phase 30
1E - Phase 31
1F - Phase 32
16 - Phase 33
21 - Phase 34
22 - Phase 35
05 - Phase 36
24 - Phase 37
25 - Phase 38
26 - Phase 39
27 - Phase 40
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0x3EC4-0x3F13 (0x50) - Game A - Pointers for Phase designs
41 BF - $BF41 (0x3F51) - Phase 03 (00)
D1 BF - $BFD1 (0x3FE1) - Phase 09 (01)
61 C0 - $C061 (0x4071) - Phase 22 (02)
F1 C0 - $C0F1 (0x4101) - Phase 23 (03)
81 C1 - $C181 (0x4191) - Phase 27 (04)
11 C2 - $C211 (0x4221) - Phase 36 (05)
A1 C2 - $C2A1 (0x42B1) - Phase 29 (06)
31 C3 - $C331 (0x4341) - Phase 05 (07)
C1 C3 - $C3C1 (0x43D1) - Phase 06 (08)
51 C4 - $C451 (0x4461) - Phase 07 (09)
E1 C4 - $C4E1 (0x44F1) - Phase 08 (0A)
71 C5 - $C571 (0x4581) - Phase 12 (0B)
01 C6 - $C601 (0x4611) - Phase 13 (0C)
91 C6 - $C691 (0x46A1) - Phase 14 (0D)
21 C7 - $C721 (0x4731) - Phase 15 (0E)
B1 C7 - $C7B1 (0x47C1) - Phase 16 (0F)
41 C8 - $C841 (0x4851) - Phase 17 (10)
D1 C8 - $C8D1 (0x48E1) - Phase 18 (11)
61 C9 - $C961 (0x4971) - Phase 19 (12)
F1 C9 - $C9F1 (0x4A01) - Phase 20 (13)
81 CA - $CA81 (0x4A91) - Phase 21 (14)
11 CB - $CB11 (0x4B21) - Phase 01 (15)
A1 CB - $CBA1 (0x4BB1) - Phase 33 (16)
31 CC - $CC31 (0x4C41) - Phase 24 (17)
C1 CC - $CCC1 (0x4CD1) - Phase 25 (18)
51 CD - $CD51 (0x4D61) - Phase 26 (19)
E1 CD - $CDE1 (0x4DF1) - Phase 04 (1A)
71 CE - $CE71 (0x4E81) - Phase 28 (1B)
01 CF - $CF01 (0x4F11) - Phase 10 (1C)
91 CF - $CF91 (0x4FA1) - Phase 30 (1D)
21 D0 - $D021 (0x5031) - Phase 31 (1E)
B1 D0 - $D0B1 (0x50C1) - Phase 32 (1F)
41 D1 - $D141 (0x5151) - Phase 02 (20)
D1 D1 - $D1D1 (0x51E1) - Phase 34 (21)
61 D2 - $D261 (0x5271) - Phase 35 (22)
F1 D2 - $D2F1 (0x5301) - Phase 11 (23)
81 D3 - $D381 (0x5391) - Phase 37 (24)
11 D4 - $D411 (0x5421) - Phase 38 (25)
A1 D4 - $D4A1 (0x54B1) - Phase 39 (26)
31 D5 - $D531 (0x5541) - Phase 40 (27)
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TSA Data
There are 4 tile values per block. The order is laid out as follows:
First value = Top-left corner tile of block
Second value = Top-right corner tile of block
Third value = Bottom-left corner tile of block
Fourth value = Bottom-right corner tile of block
0x3F14-0x3F3F (0x2C)
30 30 30 30 - 00: Black background tile
5C 5E 5D 5F - 01: Horizontal construction block
6C 6E 6D 6F - 02: Vertical construction block
36 38 37 39 - 03: Left connector of gate
40 42 41 43 - 04: Right connector of gate
34 34 35 35 - 05: Ground/bricks
50 34 50 34 - 06: Brick walls
44 47 45 48 - 07: Top of blue gate
4A 4B 3C 3F - 08: Bottom of blue gate
45 48 45 48 - 09: Middle of blue gate
30 60 30 60 - 0A: Rope
Other blocks:
0B: Blue gate open
0C: Red gate open
0D: Top of red gate
0E: Bottom of red gate
0F: Middle of red gate
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A level's object data is stored at $0300 in RAM.
Let's look at Phase 01.
Each row is 0x18 (in hex) bytes long. 24 (in decimal) blocks to draw.
There are 12 rows. 24 x 12 = 288 blocks.
Phase 01's object data in RAM is $0300-$041F (0x120).
01 01 00 00 07 00 00 0A 00 00 00 00 0A 00 00 00 07 00 00 00 00 0A 01 01
00 02 00 00 09 00 00 0A 00 00 00 00 0A 00 00 00 09 00 00 00 00 0A 02 00
00 01 01 03 08 04 01 00 01 01 07 01 0A 00 00 00 09 00 00 00 00 0A 02 00
00 02 00 00 00 0A 01 01 01 01 09 01 00 01 01 03 08 04 01 01 01 0A 02 00
00 02 00 00 00 0A 00 0A 00 00 08 04 01 01 0A 00 00 00 00 00 00 0A 02 00
00 02 00 01 01 0A 00 0A 00 00 00 00 00 00 0A 00 00 00 00 00 00 0A 02 00
00 02 00 00 01 01 01 0A 00 00 00 00 00 00 0A 00 00 01 01 01 01 0A 02 00
00 02 00 00 00 00 01 01 01 01 01 01 01 01 0A 01 01 01 00 00 00 0A 02 00
00 02 00 00 00 00 00 00 00 00 00 00 00 00 0A 00 00 00 00 00 00 0A 02 00
00 02 00 00 00 00 00 00 00 00 00 00 00 00 0A 00 00 00 00 00 00 0A 02 00
05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05
05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05
Phase 01's object data in the ROM is from 0x4B21-0x4BB0 (0x90).
11 00 70 0A 00 00 A0 00 D0 00 0A 11
02 00 90 0A 00 00 A0 00 F0 00 0A 20
01 13 84 10 11 71 A0 00 F0 00 0A 20
02 00 BA 11 11 91 01 13 E4 11 1A 20
02 00 BA 0A 00 84 11 A0 C0 00 0A 20
02 01 1A 0A 00 B0 00 A0 C0 00 0A 20
02 00 11 1A 00 B0 00 A0 C1 11 1A 20
02 00 00 11 11 11 11 A1 11 00 0A 20
02 00 00 00 00 00 00 A0 00 00 0A 20
02 00 00 00 00 00 00 A0 00 00 0A 20
55 55 55 55 55 55 55 55 55 55 55 55
55 55 55 55 55 55 55 55 55 55 55 55
288 blocks, 144 bytes in decimal representing object data.
So, each byte actually represents two blocks in the object data!
The first 11 represents the first two "01" horizontal construction blocks.
Next, 00 represents two empty tiles.
70 represents the top of a blue gate and then an empty tile.
Etc.