I am sure that vaguely "Medieval" technology could build a structure like you describe out of stone, and/or brick, and/or tiles, and/or concrete, and/or wood, and/or other materials.
You may have to modify some aspects of your design in order to keep other aspects, and thus may have to make some choices.
I suggest that you take a look at plans (if available) for the various buildings I mention, including looking up other sites besides the ones I link to. You may want to find out how thick their walls were or are and how large the interior spaces were or are.
The Pharos at Alexandria (280 BC)was very tall, allegedly 100 meters (328 feet) or 120 to 137 meters (393.7 to 449 feet), rivaling or possibly surpassing the Great Pyramid (145.6 meters or 481 feet), and probably had more interior space compared to the thickness of the walls.
In Rome the Mausoleum of Augustus (c.30 BC) had several concentric circular walls, higher in the center, to support a conical mound of dirt and growing plants.
Emperor Claudius (reigned AD 41-54) built a harbor and lighthouse at Ostia, the port of Rome. It has been suggested that for reasons of prestige Claudius must have built his lighthouse taller than the Pharos at Alexandria. Thus it might have been taller than the Great Pyramid.
The Colosseum (AD 72-80) in Rome. The Colosseum is sort of an inside out version of what you want, having an arena at ground level in the center and the highest walls on the outside, so one would have to turn it inside out to get the right plan. And of course the Colosseum was designed with many entrances and exits and you wouldn't want that in a "keep" if the "keep" was intended for serious defense purposes.
In Rome the Mausoleum of Hadrian, now part of Castel Sant'Angelo (134-139), like the Mausoleum of Augustus, had several concentric circular walls, higher in the center, to support conical mounds of dirt and growing plants.
The Rotunda in Thessaloniki, Greece (306) has two concentric circular walls, the taller inner one supported by arches and piers.
The Church of Santa Costanza in Rome (4th century) has two concentric circular walls, the inner one higher and supported on columns and arches.
The Church of Saint Stephen in Rome (5th century) has several concentric circular walls, the innermost and tallest supported on columns.
The Church of the holy Sepulchre in Bologna, Italy (5th century, rebuilt c. 1000) Has a taller inner twelve sided wall supported by 12 columns, and a lower outer octagonal wall.
The legendary Yongning Pagoda in Louyang, China, built in 516 but burned down in 534, was largely made of wood. It was certainly much taller than the proposed structure would be, so there there would be no structural problems building the proposed structure out of wood.
See posts 88, 89 on page 9 here:
The Yongning Pagoda was described in Record of the Buddhist Monasteries in Loyang to be 90 Zhang high and 100 Zhang with the spire, or 330 meters (1082.68 feet), but in the commentary of the Waterways Classic "only" 49 Zhang or 163 meters (534.777 feet). Archaeologist Yang Honxun who excavated its foundations believed it was about 147 meters (482.283 feet) tall.
Anyway, the structure asked about, if made of wood, would be much less extreme than the Yongning Pagoda.
The Church of San Vitale in Ravenna, Italy (526-547) has two concentric octagonal walls, the higher inner wall supported on arches and piers.
The Church of Saints Sergius and Bacchus in istanbul, Turkey (527-536) has a higher central wall supported by arches and piers, surrounded by a lower square wall.
The Dome of the Rock in Jerusalem (c.685/86-691/92) has a inner, higher, circular wall supported on columns and two outer, lower, octagonal walls.
The palatine chapel at Aachen, Germany (c.792-805) has two concentric walls, the inner one higher and supported on arches and piers.
The Old Cathedral, Bresica, Italy (c. 11th Century) has two concentric circular walls,the higher inner one supported by arches and piers.
The Rotunda of San Lorenzo, Mantua, Italy (late 11th century) has two concentric circular walls, the higher inner one supported by arches and piers.
The Garisenda and Asinelli Towers in Bologna, Italy (c.1109-1119?), are basically square stone towers, very different from the desired design, but reached heights of 60 and 97.2 meters (196 and 318 feet) with very thick walls but thin enough to have rooms inside even at ground level.
The Church of the Holy Sepulchre, Cambridge, England (c.1130) has two concentric walls in the nave, the inner one higher and supported on arches and piers.
The Temple Church, London, England (1185) has a circular section with a high inner wall supported on six columns surrounded by a lower circular outer wall.
The Liebfrauenkirche, Trier, Germany (13th century), has a complex, basically cross shaped and circular, gothic structure supported by piers or columns.
The central tower of Lincoln Cathedral, Lincoln, England, was raised to a height of 271 feet (83 meters) in 1307-1311, and a wooden spire was added on top of the tower, allegedly reaching a height of 525 feet (160 meters).
The Torre del Mangia, Siena, Italy, (1338-1348) is 334.6 feet (102 meters) tall.
The octagonal north tower of Strassbourg Cathedral, Strasbourg, France was completed in 1439, and is 466 feet (142 meters) tall. It was the tallest building in the world from 1647 to 1874.
The steeple tower of St. Mary's Church, Stralsund, Germany, built after 1495, was 495 feet (151 meters) tall, which made it the tallest structure in the world from 1549 to 1569 and from 1573 to 1647.
The central tower of Beauvais Cathedral, Beauvais, France, completed in 1569, was 502 feet (153 meters) tall making it the tallest structure in the world from 1569 to 1573.
A common early type of castle was called a "motte-and-bailey" castle. The bailey was a courtyard with buildings enclosed by a stone or wooden wall. The motte was a conical mound with a flat top. A usually circular keep of stone or wood was built on top of the motte, and the keep often had a central courtyard with a stone or wood tower in the center.
You give the total height of the rooms in your tower as 45 to 55 feet. You don't say if the ceilings should be flat wooden floors or vaulted stone, brick or other masonry, so I don't know how much they add to the height, and there may be parapets going a few feet above the roof. But your tower looks like it will be less than 75 feet in total height.
You say that say that "cost is not a factor". I suggest that you take a look at the castle of Coucy, built by the Lord of Coucy (not king, not duke, not count, but lord).
Also the Great dome of the Florence Cathedral, Florence, Italy (1420-1436), could easily contain your entire proposed tower.