For an overview of the different ways a census search can go wrong, see section B of Steve Morse's FAQ for his One-Step Web Pages, Which One-Step Census Form Should I Use?, which discusses search strategies.
Ordinarily with a search for missing persons in the census, I might suggest starting with a map. Lisa Louise Cooke's article How to find Enumeration District Maps gives a good overview of the process. But in this case, we already know where the hospital is on the map -- we are investigating missing pages, not missing people.
For descriptions of the EDs, NARA's Reference Report 302 Enumeration District (ED) Census Maps, 1880 – 1930 refers the reader to microcopy publication T1224:
___ T1224, Descriptions of Census Enumeration Districts, 1830 – 1890 and 1910 – 1950. 155 rolls. DP. Arranged numerically by census year, alphabetically by state, numerically by the supervisor’s districts, and then by enumeration district; with the exception of 1930. The 1930 descriptions are arranged alphabetically by county. This series contains the descriptions of the boundaries for enumeration districts. Rolls 4 – 90 cover the time period 1880 – 1930.
Steve Morse's Unified ED Finder for the 1910 Census gives results for Topeka of EDs 139-191. Putting the hospital in the finder gives results of 191 and 192. However, walking through ED 191, I did not find any patients; for ED 192, we already know from bgwiehle's comment that:
U.S. Censuses are available on archives.org as digitized microfilm (no index). ED 191 (archive.org/stream/13thcensus1910po457unit#page/n377/mode/1up ) is only 4 pages, same as at ancestry.com.
Furthermore, the FamilySearch Wiki article United States Census, 1910 (FamilySearch Historical Records)/Known Issues does not list any missing pages for Kansas (at the time I am writing this answer).
So my question is, were the pages filmed out of order, or skipped?
If you are searching for an individual patient, you can try a Soundex search. There is a Soundex index for Kansas for the 1910 Census, according to NARA's Guide, Using NARA's Census Microfilm Catalogs. The guide says:
Go to the catalog for the appropriate census year (1880, 1900, 1910, 1920, or 1930), then find the state, microfilm publication number, and roll containing the soundex code. Note that only the beginning and ending soundex codes contained on each roll are shown.
NARA's microfilm catalog for the 1910 Federal Population Census is online. The Kansas index is part of publication T1265 (Miracode).
Not in Kansas?
On Page 3 of the NARA's guide, below the listings for the individual states, there is an entry that says:
Indians, Prisoners, Insane, In Hospital, Late Registrants
That directs us to the very end of the census, roll 1784, after Wyoming.
Part 1 of the microfilm catalog describes the population schedules as follows:
The catalog also includes the 1910 census schedules for all states, Alaska, Hawaii, the District of Columbia, and Puerto Rico and for military ships, stations, hospitals, and personnel assigned to the Philippines. This is very likely to mean military hospitals, not civilian ones, but you could try looking there in case someone has 'helpfully' re-arranged the images.
In Ancestry's browse view, at FamilySearch, and Steve Morse's One-Step pages, there is a choice for Military and Naval Forces.
Under 3. Contents of Selected Roll on Morse's site, it says:
NARA Roll T624_1784 / FHL Roll 1375797: contains the following EDs in the order shown (left to right). Knowing this ordering will make it easier for you when you are scanning through the roll looking for the ED.
American Samoa: 1 to end || California: 1 to end || China: 1 to end || Cuba: 1 to end || Guam: 1 to end ||
Hawaii: 1 to end || Japan: 1 to end || Maine: 1 to end || Maryland: 1 to end || Massachusetts: 1 to end ||
Minnesota: 1 to end || New York: 1 to end || Nicaragua: 1 to end || Panama: 1 to end || Pennsylvania: 1 to end ||
Philippines: 1 to end || South Carolina: 1 to end || Turkey: 1 to end || US Army: 1 to end || US Navy: 1 to end ||
Virginia: 1 to end || Minnesota: 1 to end
That doesn't say anything about "late registrations".
If the pages were microfilmed out of order, that information is often in a Descriptive Pamphlet at NARA, but I haven't been able to locate one yet. The DP is often included at the beginning of microfilm rolls, but the digitization partners don't always include those frames.
Morse's viewer to view the rolls directly is at Viewing Census Images ... in One Step
Roll 1784 at the Internet Archive is here: https://archive.org/stream/thirteenthcensus1784usce#page/n0/mode/2up
Sometimes clues to what happened to missing records can be found in the correspondence files of the agency that created them. The next step in that case might be to see what correspondence of the US Census Bureau has been sent to the National Archives.
If no DP, reference report, or other NARA finding aid addresses the issue, you could try emailing NARA at email@example.com to see if they have more information which isn't online.
You could also check with the Kansas State Archives in case someone has written up a finding aid for Kansas, to see if anyone there has discovered the problem, or do a review of the genealogical publications to see if anyone has talked about missing census schedules.